CBSE Class 10 Mathematics Triangles MCQs Set 06

Question. It is given that \(\Delta ABC \sim \Delta PQR\) with \(\frac{BC}{QR} = \frac{1}{3}\). Then \(\frac{\text{ar}(\Delta PRQ)}{\text{ar}(\Delta BCA)}\) is equal to
(a) 9
(b) 3
(c) \(\frac{1}{3}\)
(d) \(\frac{1}{9}\)
Answer: A
Since, \(\Delta ABC \sim \Delta PQR\)
\(\frac{\text{ar}(\Delta PRQ)}{\text{ar}(\Delta BCA)} = \frac{QR^2}{AC^2}\)
\(= \frac{QR^2}{BC^2} = \frac{9}{1} = 9\) [Given \(\frac{BC}{QR} = \frac{1}{3} \implies \frac{QR}{BC} = \frac{3}{1}\)]

Question. The area of a right angled isosceles triangle whose hypotenuse is equal to \(270 \text{ m}\) is-
(a) \(19000 \text{ m}^2\)
(b) \(18225 \text{ m}^2\)
(c) \(17256 \text{ m}^2\)
(d) \(18325 \text{ m}^2\)
Answer: C
Hypotenuse \(= 270 \text{ m}\)
\(\text{Hypotenuse}^2 = \text{Side}^2 + \text{Side}^2 = 2 \text{Side}^2\)
\(\text{Side}^2 = (270)^2 / 2 = 72900 / 2 = 36450\)
or \(\text{side} = 190.91 \text{ m}\)
Required Area \(= 1/2 \times 190.91 \times 190.91\)
\(= 36446.6 / 2\)
\(= 18225 \text{ m}^2\) (approx)

Question. The areas of two similar triangles \(ABC\) and \(PQR\) are in the ratio \(9 : 16\). If \(BC = 4.5 \text{ cm}\), then the length of \(QR\) is
(a) \(4 \text{ cm}\)
(b) \(4.5 \text{ cm}\)
(c) \(3 \text{ cm}\)
(d) \(6 \text{ cm}\)
Answer: D
Since, \(\Delta ABC \sim \Delta PQR\)
\(\frac{\text{ar}(\Delta ABC)}{\text{ar}(\Delta PQR)} = \frac{BC^2}{QR^2}\)
\(\frac{9}{16} = \frac{(4.5)^2}{QR^2}\)
\(QR^2 = \frac{16 \times (4.5)^2}{9}\)
\(QR = 6 \text{ cm}\)

Question. If \(\Delta ABC \sim \Delta APQ\) and \(\text{ar}(\Delta APQ) = 4 \text{ ar}(\Delta ABC)\), then the ratio of \(BC\) to \(PQ\) is
(a) \(2 : 1\)
(b) \(1 : 2\)
(c) \(1 : 4\)
(d) \(4 : 1\)
Answer: B
Since, \(\Delta ABC \sim \Delta APQ\)
\(\frac{\text{ar}(\Delta ABC)}{\text{ar}(\Delta APQ)} = \frac{BC^2}{PQ^2}\)
\(\frac{\text{ar}(\Delta ABC)}{4 \cdot \text{ar}(\Delta ABC)} = \frac{BC^2}{PQ^2}\)
\(\left(\frac{BC}{PQ}\right)^2 = \frac{1}{4}\)
\(\frac{BC}{PQ} = \frac{1}{2}\)

Question. The length of the side of a square whose diagonal is \(16 \text{ cm}\), is
(a) \(8\sqrt{2} \text{ cm}\)
(b) \(2\sqrt{8} \text{ cm}\)
(c) \(4\sqrt{2} \text{ cm}\)
(d) \(2\sqrt{2} \text{ cm}\)
Answer: A

Question. \(\Delta ABC\) is an equilateral triangle with each side of length \(2p\). If \(AD \perp BC\) then the value of \(AD\) is
(a) \(\sqrt{3}\)
(b) \(\sqrt{3}p\)
(c) \(2p\)
(d) \(4p\)
Answer: B
Given an equilateral triangle \(ABC\) in which,
\(AB = BC = CA = 2p\)
and \(AD \perp BC\)
In \(\Delta ADB\), \(AB^2 = AD^2 + BD^2\) (By Pythagoras theorem)
\((2p)^2 = AD^2 + p^2\)
\(AD = \sqrt{3}p\)

Question. Which of the following statement is false?
(a) All isosceles triangles are similar.
(b) All equilateral triangles are similar.
(c) All circles are similar.
(d) None of the above
Answer: A
An isosceles triangle is a triangle with two sides of equal length hence statement given in option (a) is false.

Question. Two poles of height \(6 \text{m}\) and \(11 \text{m}\) stand vertically upright on a plane ground. If the distance between their foot is \(12 \text{ m}\), then distance between their tops is
(a) \(12 \text{ m}\)
(b) \(14 \text{ m}\)
(c) \(13 \text{ m}\)
(d) \(11 \text{ m}\)
Answer: C

FILL IN THE BLANK

Question. All congruent figures are similar but the similar figures need .......... be congruent.
Answer: not

Question. If two polygons are similar then the same ratio of the corresponding sides is referred to as the ..........
Answer: scale factor

Question. Two polygons of the same number of sides are similar, if all the corresponding angles are ..........
Answer: equal

Question. Two figures are said to be .......... if they have same shape but not necessarily the same size.
Answer: similar

Question. All circles are ..........
Answer: similar

Question. .......... theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Answer: Basic proportionality

Question. All .......... triangles are similar.
Answer: equilateral

Question. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the .......... ratio.
Answer: same

Question. Two figures having the same shape and size are said to be ...........
Answer: congruent

Question. The diagonals of a quadrilateral \(ABCD\) intersect each other at the point \(O\) such that \(\frac{AO}{BO} = \frac{CO}{DO}\). \(ABCD\) is a ..........
Answer: trapezium

TRUE/FALSE

Question. A \(\Delta ABC\) with \(AB = 24\) cm, \(BC = 10\) cm and \(AC = 26\) cm is a right triangle.
Answer: True

Question. Two figures having the same shape but not necessarily the same size are called similar figures.
Answer: True

Question. If \(\Delta DEF \sim \Delta QRP\), then \(\angle D = \angle Q\) and \(\angle E = \angle P\).
Answer: False

Question. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar.
Answer: True

Question. In a right triangle, the square of the hypotenuse is equal to the sum of the other two sides.
Answer: False

Question. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Answer: True

Question. If \(\Delta ABC \sim \Delta XYZ\), than \(\frac{AB}{XY} = \frac{AC}{XZ}\).
Answer: True

Question. If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
Answer: True

Question. If \(\Delta DEF \sim \Delta PQR\), \(ar(\Delta DEF) = 9\) sq. units, then \(ar(\Delta PQR) : ar(\Delta DEF) = 4 : 3\).
Answer: True

Question. Diagonals \(AC\) and \(BD\) of a trapezium \(ABCD\) with \(AB \parallel DC\) intersect each other at the point \(O\), \(\frac{OA}{OC} = \frac{OB}{OD}\).
Answer: True

ASSERTION AND REASON

DIRECTION: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : If in a \(\Delta ABC\), a line \(DE \parallel BC\), intersects \(AB\) in \(D\) and \(AC\) in \(E\), then \(\frac{AB}{AD} = \frac{AC}{AE}\).
Reason : If a line is drawn parallel to one side of a triangle intersecting the other two sides, then the other two sides are divided in the same ratio.
(a) A
(b) B
(c) C
(d) D
Answer: A

Question. Assertion : In \(\Delta ABC\), \(DE \parallel BC\) such that \(AD = (7x - 4)\) cm, \(AE = (5x - 2)\) cm, \(DB = (3x + 4)\) cm and \(EC = 3x\) cm than \(x\) equal to 5.
Reason : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distant point, than the other two sides are divided in the same ratio.
(a) A
(b) B
(c) C
(d) D
Answer: D
Assertion (A) is false but reason (R) is true. We have, \(\frac{AD}{DB} = \frac{AE}{EC} \Rightarrow \frac{7x-4}{3x+4} = \frac{5x-2}{3x} \Rightarrow 21x^2 - 12x = 15x^2 + 20x - 6x - 8 \Rightarrow 6x^2 - 26x + 8 = 0 \Rightarrow 3x^2 - 13x + 4 = 0 \Rightarrow 3x^2 - 12x - x + 4 = 0 \Rightarrow 3x(x - 4) - 1(x - 4) = 0 \Rightarrow (x - 4)(3x - 1) = 0 \Rightarrow x = 4, \frac{1}{3}\). So, A is incorrect but R is correct.

Question. Assertion : \(\Delta ABC \sim \Delta DEF\) such that \(ar(\Delta ABC) = 36 \text{ cm}^2\) and \(ar(\Delta DEF) = 49 \text{ cm}^2\) then, \(AB : DE = 6 : 7\).
Reason : If \(\Delta ABC \sim \Delta DEF\), then \(\frac{ar(\Delta ABC)}{ar(\Delta DEF)} = \frac{AB^2}{DE^2} = \frac{BC^2}{EF^2} = \frac{AC^2}{DF^2}\).
(a) A
(b) B
(c) C
(d) D
Answer: A
Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A). \(\frac{ar(\Delta ABC)}{ar(\Delta DEF)} = \frac{AB^2}{DE^2} \Rightarrow \frac{36}{49} = \frac{AB^2}{DE^2} \Rightarrow \frac{AB}{DE} = \frac{6}{7}\). \(AB:DE = 6:7\). So, both A and R are correct and R explain A.

Question. Assertion : \(\Delta ABC\) is an isosceles triangle right angled of \(C\), then \(AB^2 = 2AC^2\).
Reason : In right \(\Delta ABC\), right angled at \(B\), \(AC^2 = AB^2 + BC^2\).
(a) A
(b) B
(c) C
(d) D
Answer: A
Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A). In an isosceles \(\Delta ABC\), right angled at \(C\) is \(AB^2 = AC^2 + BC^2 \Rightarrow AB^2 = AC^2 + AC^2 \Rightarrow AB^2 = 2AC^2\) (as \(AC = BC\)). So, both A and R are correct and R explains A.