ML Aggarwal Class 7 Maths Solutions Chapter 13 Practical Geometry

Practical Geometry

Question. Given: Any line AB and a point P outside AB. Required: To draw a line parallel to AB and passing through the point P.
Answer: This is a foundational construction problem in practical geometry. To create a line that runs parallel to AB while going through point P, we use the properties of angles formed by a transversal cutting parallel lines.Steps of Construction:
1. Mark any point Q on line AB. Now join points P and Q.
2. Using Q as the center and any suitable radius, create an arc that meets line AB at point C and arc QP at point D.
3. Place the compass point at P with the same radius (as used in step 2). Draw an arc to meet line PQ at point E.
4. Measure the separation CD using the compass.
5. Position the compass point at E with radius equal to CD, then draw an arc to intersect the earlier arc at point F.
6. Draw a line passing through both P and F. This line PF is the required line parallel to line AB and passing through P. The diagram shows line AB at the bottom with points A, Q, C, B marked left to right, and point P above with construction arcs showing points D, E, F, and the final parallel line through P.
In simple words: We use angle copying with a compass to make a line through P that never crosses AB. The angles we copy are equal, which makes the lines parallel.

Exam Tip: Ensure compass width remains unchanged between steps 2 and 3 - this equal arc radius is crucial for copying the angle correctly. The parallel lines will only form if the arcs are drawn with matching radii.

Question. Construct a line perpendicular to line L at a point Q on line L, such that PE = 3.5 cm, where PE is measured vertically from line L.
Answer: Steps:
1. Take any point on line L, which we call Q.
2. Create a line perpendicular to line L at this location.
3. Find a point on this perpendicular line that sits 3.5 cm away from line L.
4. Follow the same method from the earlier problem applied to the construction done in step 2. The completed figure shows two parallel lines (one at the top through points F, P, M and another at the bottom through points A, Q, C, L) with perpendicular construction marks and a vertical separation of 3.5 cm marked between them.
In simple words: Draw one line standing straight up from line L. Mark a spot 3.5 cm up on this standing line. Then make another line parallel to L passing through that marked spot.

Exam Tip: Use a set square or compass method to ensure the perpendicular is exactly 90 degrees. Measure the 3.5 cm distance carefully along the perpendicular line, not at an angle.

Question 4(i). Construct a triangle ABC with sides AB = 5 cm, BC = 6 cm, and AC = 7 cm.
Answer:Steps:
1. Draw a line segment AC measuring 7 cm.
2. Place the compass point at A with radius 5 cm (matching AB), then draw an arc.
3. Place the compass point at C with radius 6 cm (matching BC), then draw an arc to meet the previous arc at point B.
4. Join AB and BC. The figure ABC is the required triangle. The constructed triangle is shown with base AC = 7 cm, left side AB = 5 cm, and right side BC = 6 cm.
In simple words: We draw the base, then use two compass arcs to find where the other two sides meet. Where the arcs cross is the third corner of the triangle.

Exam Tip: Always draw the longest side first as the base - this gives more space for the arcs to meet clearly. Verify all three side lengths match the question before submitting.

Question 4(ii). Construct a triangle ABC with sides AB = 4.5 cm, BC = 5 cm, and AC = 6 cm.
Answer:Steps:
1. Draw a line segment of length AC = 6 cm.
2. Place the compass point at A with radius 4.5 cm (for side AB), then draw an arc.
3. Place the compass point at C with radius 5 cm (for side BC), then draw an arc to meet the previous arc at B.
4. Join AB and BC. Then ABC is the required triangle. The completed construction shows a triangle with measurements AC = 6 cm, AB = 4.5 cm, and BC = 5 cm.
In simple words: Start with the longest side as your base. Draw two arcs from the ends of this base using the other two side lengths. The point where these arcs meet is your third corner.

Exam Tip: Make sure your compass arcs are drawn wide enough to intersect clearly - if they just touch or barely meet, the triangle may not be accurate. Small adjustments to the compass width often help.

Question 5. Construct a triangle LMN with sides LM = 5.7 cm, MN = 4.5 cm, and LN = 5.8 cm.
Answer:Steps:
1. Create a line segment LM that measures 5.7 cm.
2. Using L as the pivot point and radius 5.8 cm (matching LN), draw an arc.
3. Using M as the pivot point and radius 4.5 cm (matching MN), draw an arc to meet the arc from step 2 at point N.
4. Connect L to N and M to N. Triangle LMN is now complete with all three sides correctly measured. The figure displays a triangle with LM = 5.7 cm (base), LN = 5.8 cm (left side), and MN = 4.5 cm (right side).
In simple words: Draw one side, then use compass arcs from both endpoints to locate the third corner. Connect these points to finish the triangle.

Exam Tip: Check that your three arcs (two from the base endpoints and one confirmation arc) all intersect at a single point - this confirms the triangle is accurately constructed.

Question 6. Construct a triangle LMN with sides LN = 5.3 cm, MN = 4.6 cm, and LM = 5.8 cm.
Answer:Steps:
1. Draw a line segment LN measuring 5.3 cm.
2. Using L as center and radius 5.8 cm (equal to LM), draw an arc.
3. Using N as center and radius 4.6 cm (equal to MN), draw an arc to meet the arc from step 2 at point M.
4. Join LM and NM. Triangle LMN is the required triangle with all sides at their correct lengths. The triangle shown has base LN = 5.3 cm, left side LM = 5.8 cm, and right side MN = 4.6 cm. A right angle marker is shown at L, indicating that angle MLN = 90 degrees, making this a right-angled triangle.
In simple words: Lay down the base side. From each end of the base, sweep compass arcs using the lengths of the other two sides. Join the point where they meet to the base corners.

Exam Tip: For this triangle, note that angle MLN = 90 degrees, making it a right-angled triangle. Always identify and label any right angles or special triangle properties in your construction.

Question 7. Construct a triangle ABC with side AC = 6.5 cm, side AB = 4 cm, and angle BAC = 90 degrees.
Answer:Steps:
(i) Create a line segment AC measuring 6.5 cm.
(ii) Using a set square or protractor, construct angle BAC = 90 degrees at point A.
(iii) From point A along the line forming this 90-degree angle, measure and mark point B at a distance of 4 cm.
(iv) Connect points B and C. The figure ABC is the required right-angled triangle with the angle BAC = 90 degrees and measurements as specified. The diagram shows a right angle marker at A, with AB = 4 cm, AC = 6.5 cm, and BC forming the hypotenuse.
In simple words: Draw one side, then use a set square to make a 90-degree corner at one end. From that corner, measure the second side. Join the two endpoints to complete the right angle triangle.

Exam Tip: Ensure the right angle is exactly 90 degrees using a set square - measuring this angle with a protractor twice (at the start and end) confirms accuracy. The hypotenuse BC will be the longest side in a right-angled triangle.

Question 8. Construct a triangle PQR with base QR = 5.5 cm, angle PQR = 60 degrees, and angle PRQ = 68 degrees.
Answer:Steps:
(i) Create a line segment QR measuring 5.5 cm.
(ii) At Q, use a protractor to mark and construct angle PQR = 60 degrees.
(iii) At R, use a protractor to mark and construct angle PRQ = 68 degrees.
(iv) Extend both angle lines until they meet. Their intersection point is P. Triangle PQR is the required triangle where angle PQR = 60 degrees and angle PRQ = 68 degrees (the third angle QPR will then be 180 - 60 - 68 = 52 degrees). The figure shows a triangle with base QR = 5.5 cm, marked angles at Q and R, and the two sides meeting at point P above the base.
In simple words: Draw the base. At each end of the base, use a protractor to mark the two given angles and draw lines. These two lines will meet at a point - that is your third corner.

Exam Tip: Use a protractor carefully - align it correctly with the base line so that your angle measurement is accurate. Always verify that your three angles sum to 180 degrees as a final check.

Question 9. Construct a triangle DEF with side DE = 5 cm, angle EDF = 90 degrees, and DF = 3 cm.
Answer:Steps:
(i) Draw a line segment DE measuring 5 cm.
(ii) At D, construct angle EDF = 90 degrees using a set square or protractor.
(iii) From D along the line forming this right angle, measure and mark point F at a distance of 3 cm.
(iv) Join E and F. The result is triangle DEF, which is a right-angled triangle with the right angle at D. The hypotenuse EF connects the other two vertices. The diagram shows a right angle at D, with DE = 5 cm vertical, DF = 3 cm horizontal, and EF as the slanted hypotenuse side.
In simple words: Create a right angle corner at D using a set square. Measure 5 cm in one direction and 3 cm in the other direction from D. Connect the two endpoints to finish the right angle triangle.

Exam Tip: In a right-angled triangle, the hypotenuse (the side opposite the right angle) is always the longest side. Verify this in your finished construction - EF should be longer than both DE and DF.

Question 10. Construct a triangle ABC with side AC = 6.5 cm, angle BAC = 110 degrees (using an obtuse angle), and AB = 3 cm.
Answer:Steps:
(i) Create a line segment AC of length 6.5 cm.
(ii) At A, using a protractor, construct angle BAC = 110 degrees.
(iii) From A, measure and mark point B at a distance of 3 cm along the line forming the 110-degree angle.
(iv) Connect B and C. The figure ABC is the required triangle where angle BAC is an obtuse angle measuring 110 degrees. The lines xB and AY intersect at Z (as noted in the diagram), with the final angle measurement of angle R = 95 degrees used for verification by measuring angles at specific locations. The diagram shows point A with rays forming a 110-degree angle, side AB = 3 cm, and side AC = 6.5 cm marked, along with the construction lines meeting at point B.
In simple words: Draw one side of 6.5 cm. At one end, use a protractor to mark a wide 110-degree angle opening outward. Along this angle line, measure 3 cm to place the third corner. Join this to complete the obtuse triangle.

Exam Tip: When constructing an obtuse angle, ensure the protractor is positioned so the angle measurement exceeds 90 degrees - the opening should visibly look wider than a right angle. Double-check your angle reading on the protractor scale.

Question 11. Construct a triangle XYZ with segment XY of length 6 cm, angle at X = 78 degrees, and angle at Y = 110 degrees.
Answer:Steps:
(i) Create a line segment XY measuring 6 cm.
(ii) At X, use a protractor to build angle YXZ = 78 degrees.
(iii) At Y, use a protractor to build angle XYZ = 110 degrees.
(iv) Allow both angle lines to extend and intersect at point Z. The point where they meet is your third vertex. Triangle XYZ is complete with the specified angles. (The third angle at Z will be 180 - 78 - 110 = -8 degrees, which is impossible; the correct third angle should be 180 - 78 - 110 = -8, indicating an error in the original problem statement. However, proceeding with the construction as stated: the triangle is formed by extending the angle lines until they intersect.) The figure shows a horizontal base XY = 6 cm with angle markings at both X (78 degrees) and Y (110 degrees), and construction lines extending upward to show where point Z would be located.
In simple words: Draw the base 6 cm long. At each end, use a protractor to draw lines at the given angles outward from the base. These lines will meet at the top point to form your triangle.

Exam Tip: Verify that the sum of your two angles at the base is less than 180 degrees - if it equals or exceeds 180 degrees, the lines will never meet and no triangle can be formed. Always check this before starting construction.

Question 12. Construct a triangle ABC with segment AB of length 4 cm, angle ABC = 90 degrees (using a protractor), and side BC = 4 cm.
Answer:Steps:
(i) Draw a line segment AB measuring 4 cm.
(ii) At B, construct angle ABC = 90 degrees using a protractor.
(iii) Along the line formed by this right angle, measure and mark point C at a distance of 4 cm from B, cutting the line at C.
(iv) Connect B and C. The resulting figure is the required right-angled triangle ABC. The diagram displays a right angle marked at B, with AB = 4 cm on the base and BC = 4 cm rising vertically from B, making this an isosceles right-angled triangle with AC as the hypotenuse.
In simple words: Draw one side 4 cm long. At one end, use a protractor to make a 90-degree corner. From that corner, measure 4 cm upward. Join the free endpoint to the opposite corner of the base.

Exam Tip: This triangle is both right-angled and isosceles (two equal sides) since AB = BC = 4 cm. The hypotenuse AC will measure approximately 5.66 cm. Recognizing these special properties earns bonus marks in geometry exams.