Solution: False According to the given question, Let’s assume that, $A\text{ }=\text{ }25\text{ }cm$ $B\text{ }=\text{ }5\text{ }cm$ $C\text{ }=\text{ }24\text{ }cm$ Using the Pythagoras Theorem, We...
In ΔPQR and ΔMST, ∠P = 55°, ∠Q =25°, ∠M = 100° and ∠S = 25°. Is ΔQPR ~ ΔTSM? Why?
Solution: We all know that, When the three angles of a triangle are added then their sum equals to 180°. Then, from triangle PQR, $\angle P\text{ }+~\angle Q\text{ }+~\angle R\text{ }=\text{...
In figure, BD and CE intersect each other at the point P. Is ΔPBC ~ ΔPDE? Why?
Solution: True In triangles PBC and PDE, ∠EPD = ∠BPC [ as vertically opposite angles] $PB/PD\text{ }=\text{ }5/10\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}$… (i) $PC/PE\text{ }=\text{...
A and B are respectively the points on the sides PQ and PR of a ΔPQR such that PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm. Is AB || QR? Give reason for your answer.
Solution: True According to the given question, $PQ\text{ }=\text{ }12.5\text{ }cm$ $PA\text{ }=\text{ }5\text{ }cm$ $BR\text{ }=\text{ }6\text{ }cm$ $PB\text{ }=\text{ }4\text{ }cm$ Then, $QA\text{...
In figure, two line segments AC and BD intersect each other at the point P such that PA = 6 cm, PB = 3 cm, PC = 2.5 cm, PD = 5 cm, ∠APB = 50° and ∠CDP = 30°. Then, ∠PBA is equal to (a) 50° (b) 30° (c) 60° (d) 100°
Solution: (d) 100° Explanation: From triangles APB and CPD, $\angle APB\text{ }=~\angle CPD\text{ }=\text{ }50{}^\circ $ (as they are vertically opposite angles) $AP/PD\text{ }=\text{ }6/5$ … (i)...
If in two Δ PQR, AB/QR = BC/PR = CA/PQ, then (a)Δ PQR~Δ CAB (b) Δ PQR ~ Δ ABC (c)Δ CBA ~ Δ PQR (d) Δ BCA ~ Δ PQR
Solution: (a)Δ PQR~Δ CAB Explanation: From triangles ABC and PQR, we have, AB/QR = BC/PR = CA/PQ When the sides of one triangle are proportional to the sides of the other given triangle, and even...
If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true? (a) BC · EF = AC · FD (b) AB · EF = AC · DE (c) BC · DE = AB · EF (d) BC · DE = AB · FD
Solution: (c) BC · DE = AB · EF Explanation: We all know that, If the sides of one triangle are proportionate to the sides of the other triangle, and the corresponding angles are all equal, the...
If the lengths of the diagonals of rhombus are 16 cm and 12 cm. Then, the length of the sides of the rhombus is (a) 9 cm (b) 10 cm (c) 8 cm (d) 20 cm
Solution: (b) 10 cm Explanation: We all know that, A rhombus is a simple quadrilateral with four equal-length sides and diagonals that are perpendicular bisector of each other. Now according to the...
1. In a Δ ABC, AD is the bisector of ∠ A, meeting side BC at D. (vii) if \[\mathbf{AB}\text{ }=\text{ }\mathbf{5}.\mathbf{6}\text{ }\mathbf{cm},\] \[\mathbf{BC}\text{ }=\text{ }\mathbf{6}\text{ }\mathbf{cm},\] and \[\mathbf{BD}\text{ }=\text{ }\mathbf{3}.\mathbf{2}\text{ }\mathbf{cm},\] find AC. (viii) if \[\mathbf{AB}\text{ }=\text{ }\mathbf{10}\text{ }\mathbf{cm},\] \[\mathbf{AC}\text{ }=\text{ }\mathbf{6}\text{ }\mathbf{cm},\] and \[\mathbf{BC}\text{ }=\text{ }\mathbf{12}\text{ }\mathbf{cm},\] find BD and DC.
Solution: Given: Δ ABC and AD bisects ∠A, meeting side BC at D. \[AB\text{ }=\text{ }5.6\text{ }cm,\] \[BC\text{ }=\text{ }6\text{ }cm,\] and \[BD\text{ }=\text{ }3.2\text{ }cm\]....
14. The lengths of the diagonals of a rhombus is \[\mathbf{24cm}\text{ }\mathbf{and}\text{ }\mathbf{10cm}.\]Find each side of the rhombus.
Solution: Let ABCD be a rhombus and AC and BD be the diagonals of ABCD. So, AC = \[24cm\text{ }and\text{ }BD\text{ }=\text{ }10cm\] \[\] We know that diagonals of a rhombus bisect each other at...
13. In a ∆ABC, AB = BC = CA = \[\mathbf{2a}\] and AD ⊥ BC. Prove that
(i) AD = a\[\surd \mathbf{3}\] (ii) Area (∆ABC) = \[\surd \mathbf{3}\text{ }{{\mathbf{a}}^{\mathbf{2}}}\] Solution: (i) In ∆ABD and ∆ACD, we have∠ADB = ∠ADC = \[{{90}^{o}}\]AB = AC [Given]AD =...
12. In an isosceles triangle ABC, if AB = AC = \[\mathbf{13cm}\] and the altitude from A on BC is
\[\mathbf{5cm},\]find BC. Solution: Given, An isosceles triangle ABC, AB = AC = \[13cm,\text{ }AD\text{ }=\text{ }5cm\] Required to find: BC In ∆ ADB, by using Pythagoras theorem, we have...
11. ABCD is a square. F is the mid-point of AB. BE is one third of BC. If the area of ∆ FBE =\[\mathbf{108c}{{\mathbf{m}}^{\mathbf{2}}}\], find the length of AC.
Solution: Given, ABCD is a square. And, F is the mid-point of AB. BE is one third of BC. Area of ∆ FBE = \[108c{{m}^{2}}\] Required to find: length of AC Let’s assume the sides of the square to be...
10. A triangle has sides \[\mathbf{5}\text{ }\mathbf{cm},\text{ }\mathbf{12}\text{ }\mathbf{cm}\text{ }\mathbf{and}\text{ }\mathbf{13}\text{ }\mathbf{cm}.\]Find the length to one decimal place, of the perpendicular from the opposite vertex to the side whose length is \[\mathbf{13}\text{ }\mathbf{cm}.\]
Solution: From the fig. \[AB\text{ }=\text{ }5cm,\text{ }BC\text{ }=\text{ }12\text{ }cm\text{ }and\text{ }AC\text{ }=\text{ }13\text{ }cm.\] Then, \[A{{C}^{2}}~=\text{ }A{{B}^{2}}~+\text{...
9. Using Pythagoras theorem determine the length of AD in terms of b and c shown in Fig\[.\text{ }\mathbf{4}.\mathbf{219}\]
Solution: We have, In ∆BAC, by Pythagoras theorem, we have \[\begin{array}{*{35}{l}} B{{C}^{2}}~=\text{ }A{{B}^{2}}~+\text{ }A{{C}^{2}} \\ \Rightarrow...
8. Two poles of height \[\mathbf{9}\text{ }\mathbf{in}\text{ }\mathbf{and}\text{ }\mathbf{14}\text{ }\mathbf{m}\] stand on a plane ground. If the distance between their feet is \[\mathbf{12}\text{ }\mathbf{m},\]find the distance between their tops.
Solution: Comparing with the figure, it’s given that AC = \[14\text{ }m,\text{ }DC\text{ }=\text{ }12m\text{ }and\text{ }ED\text{ }=\text{ }BC\text{ }=\text{ }9\text{ }m\] Construction: Draw...
7. The foot of a ladder is \[\mathbf{6}\text{ }\mathbf{m}\] away from a wall and its top reaches a window \[\mathbf{8}\text{ }\mathbf{m}\] above the ground. If the ladder is shifted in such a way that its foot is \[\mathbf{8}\text{ }\mathbf{m}\] away from the wall, to what height does its tip reach?
Solution: Let’s assume the length of ladder to be, AD = BE = x m So, in ∆ACD, by Pythagoras theorem We have, \[\begin{array}{*{35}{l}} ...
6. In an isosceles triangle ABC, AB = \[\mathbf{AC}\text{ }=\text{ }\mathbf{25}\text{ }\mathbf{cm},\text{ }\mathbf{BC}\text{ }=\text{ }\mathbf{14}\text{ }\mathbf{cm}.\]Calculate the altitude from A on BC.
Solution: Given, ∆ABC, AB = AC = \[25\text{ }cm\text{ }and\text{ }BC\text{ }=\text{ }14.\] \[\] In ∆ABD and ∆ACD, we see...
5. Two poles of heights \[\mathbf{6}\text{ }\mathbf{m}\text{ }\mathbf{and}\text{ }\mathbf{11}\text{ }\mathbf{m}\]stand on a plane ground. If the distance between their feet is\[\mathbf{12}\text{ }\mathbf{m}\], find the distance between their tops.
Solution: Let CD and AB be the poles of height \[11m\text{ }and\text{ }6m.\] Then, its seen that CP = \[11\text{ }\text{ }6\text{ }=\text{ }5m.\] From the figure, AP should be \[12m\] (given) In...
4. A ladder \[\mathbf{17}\text{ }\mathbf{m}\]long reaches a window of a building \[\mathbf{15}\text{ }\mathbf{m}\]above the ground. Find the distance of the foot of the ladder from the building.
Solution: In ∆ABC, by Pythagoras theorem \[\begin{array}{*{35}{l}} A{{B}^{2}}~+\text{ }B{{C}^{2}}~=\text{ }A{{C}^{2}} \\ \Rightarrow {{15}^{2}}~+\text{...
3. A man goes \[\mathbf{15}\text{ }\mathbf{metres}\]due west and then \[\mathbf{8}\text{ }\mathbf{metres}\]due north. How far is he from the starting point?
Solution:  ...
2. The sides of certain triangles are given below. Determine which of them are right triangles.
\[\begin{array}{*{35}{l}} \left( \mathbf{i} \right)\text{ }\mathbf{a}\text{ }=\text{ }\mathbf{7}\text{ }\mathbf{cm},\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{24}\text{...
1. If the sides of a triangle are \[\mathbf{3}\text{ }\mathbf{cm},\text{ }\mathbf{4}\text{ }\mathbf{cm},\text{ }\mathbf{and}\text{ }\mathbf{6}\text{ }\mathbf{cm}\]long, determine whether the triangle is a right-angled triangle.
Solution: We have, Sides of triangle as \[\begin{array}{*{35}{l}} AB\text{ }=\text{ }3\text{ }cm \\ BC\text{ }=\text{ }4\text{ }cm \\ AC\text{...