Perpendicular from Right Angle to Hypotenuse Divides the Triangle into Two Similar Triangles

Q.

If $∆ABC~∆QRP$, $\frac{ar(∆ABC)}{ar(∆PQR)}=\frac{9}{4}$, $AB=18cm$ and $BC=15cm$, then find the value of $PR$.

Q.

In the given figure, if $\angle ACB=\angle CDA,AC=8cm$ and $AD=3cm$, then find $BD$.

Q. ABC is an equilateral triangle of side 4a. Length of its altitude is .

1. $2\sqrt{3}a$
2. $7\sqrt{9}a$
3. $4\sqrt{3}a$

Q.

In the adjoining figure, if BC = a, AC = b, AB = c and $\angle CAB={120}^{\circ }$, then the correct relation is-

1. a² = b² + c² – 2 bc

2. a² = b² + c² + bc

3. a² = b² + c² + 2 bc

4. a² = b² + c² – bc

Q.

$△$ABC is a right angled triangle, right angled at B. BD is perpendicular to AC. What is AC . DC?

1. ${BC}^2$

2. ${BD}^2$

3. BC.AB

4. AB.AC

Q.

In the figure $△ABC$ is a right angled triangle with right angle at B. BD is perpendicular to AC. Then which of the following options will hold true?

1. AB² = AD × DC

2. AB² = AD × AC

3. AD² = DC × AC

4. AB² = DC² + AD²

Q.

Two sides of an isosceles triangle measure $5cmand13cm.$ What is the possible values of third side?

Q.

Prove that the angle between the internal bisector of one base angle and the external bisector of the other is equal to one half of the vertical angle.

Q. D, E and F are the points on sides BC, CA and AB respectively of ∆ABC such that AD bisects ∠A, BE bisects ∠B and CF bisects ∠C. If AB = 5 cm, BC = 8 cm and CA = 4 cm, determine AF, CE and BD.

Q.

Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

Q. $Inthe\Delta ABC,DE||BCand\frac{AD}{DB}=\frac{2}{5}.$ If $AC=5.6cm$, then $AE=$____

1. 1.6 cm
2. 4 cm
3. 3.5 cm
4. 3 cm

Q.

$△$ABC is a right angled triangle, right angled at B. BD is perpendicular to AC. What is AC . DC?

1. $BC.AB$

2. ${BC}^2$

3. ${BD}^2$

4. $AB.AC$

Q. Find the value of $x$ in the following figure.

1. 5cm
2. 3cm
3. 4.8cm
4. 5.2cm

Q.

In a right angled triangle ABC in which ∠A = ${90}^{\circ }$, If AD $\perp$ BC.

1. ${AB}^2$ = BD × DC.

2. ${AB}^2$ = BD × AD.

3. ${AB}^2$ = BC × DC.

4. ${AB}^2$ = BC × BD.

Q.

Two sides of an isosceles triangle measure $4cmand11cm.$ What is the possible value of third side.

Q.

Question 62

Observe all the four triangles FAB, EAB, DAB and CAB as shown in the figure.

State whether the statement is True or False.

All triangles have the same base and the same altitude.

Q.

The measure of each of two angles of a triangle is $70°.$ The measure of a third angle is$\_\_\_$

Q.

Name the types of following triangles:

(a) Triangle with lengths of sides 7 cm, 8 cm and 9 cm.

(b) ΔABC with AB = 8.7 cm, AC = 7 cm and BC = 6 cm.

(c) ΔPQR such that PQ = QR = PR = 5 cm.

(d) ΔDEF with m∠D = 90°

(e) ΔXYZ with m∠Y = 90° and XY = YZ.

(f) ΔLMN with m∠L = 30°, m∠M = 70° and m∠N = 80°

Q. $△ABC$ is right angled at B and the perpendicular drawn from B to the opposite side AC bisects it at D. If AD = DC = 5 cm, then find the length of BD.

1. 5 cm
2. 10 cm
3. 25 cm
4. 12.5 cm

Q.

BO and CO are respectively the bisectors of $\angle B$ and $\angle C$ of $\Delta ABC$. AO produced meets BC at P.


Show that [4 MARKS]

(i) $\frac{AB}{BP}=\frac{AO}{OP}$

(ii) $\frac{AC}{CP}=\frac{AO}{OP}$

(iii) $\frac{AB}{AC}=\frac{BP}{PC}$

(iv) $AP$ is the bisector of $\angle BAC$.

Q. Length of the altitude on the hypotenuse of a right angled triangle with area $A{\text{cm}}^2$ and one of the sides equal to $b\text{cm}$ is __ cm.

1. $\frac{2A{b}^2}{\sqrt{b^4+4A^2}}$
2. $\frac{2b}{\sqrt{b^4+4A^2}}$
3. $\frac{2b^{2}A}{\sqrt{b^4+A^2}}$
4. $\frac{2bA}{\sqrt{b^4+4A^2}}$

Q.

key points of chapter triangles and the theorems

Q. Find the correct option.

1. ∆ABC $\sim$ ∆QPR
2. ∆ABC $\sim$ ∆PQR
3. ∆ABC $\sim$ ∆RPQ
4. ∆ABC $\sim$ ∆RQP

Q. ABC is an equilateral triangle. If D is a point on side BC such that BD : BC =1:3, then $A{D}^2:A{B}^2$ = .

1. 7 : 9
2. 4 : 3
3. 11 : 9
4. 17 : 3

Q. Find the value of $BD\times DC$ , if $AD=5cm$.

1. $25c{m}^2$
2. $10c{m}^2$
3. $12c{m}^2$
4. $15c{m}^2$

Q. Question 10 (i)
CD and GH are respectively the bisectors of $\angle ACBand\angle EGF$ such that D and H lie on sides AB and FE of $\Delta ABC$ and $\Delta EFG$ respectively. If $\Delta ABC\sim \Delta FEG$, Show that
$\left(i\right)\frac{CD}{GH}=\frac{AC}{FG}$

Q.

A perpendicular drawn from the vertex of one of the acute angles of a right angled triangle divides the triangle into two triangles similar to each other.

1. True

2. False

Q. Question 3 (ii)
In the given figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that:
(ii) $A{C}^2=BC\times DC$

Q. Question 4
In the figure, BD and CE intersect each other at the point P. Is $\Delta PBC\sim \Delta PDE$? Why?

Q.

In $△$ABC, B is a right angle, and BD is perpendicular to AC and DE is perpendicular to BC. Then $AD.DC=$

1. $B{D}^2$

2. $BE.BC$

3. $AB.BC$

4. $BD.DC$