Reducing a Triangle to Two Right Angled Triangles

Q. If the distance between the points (3, 0) and (0, y) is 5 units and y is positive. then what is the value of y?

Q. If the points(x, 4) lies on a circle whose centre is at the origin and radius is 5, then x =

(a) ±5

(b) ±3

(c) 0

(d) ±4

Q. In a right angled triangle with sides a and b and hypotenuse c, the altitude drawn on the hypotenuse is x. Prove that ab = cx.

Q. If the point P(2, 2) is equidistant from the points A(−2, k) and B(−2k, −3), find k. Also find the length of AP.

Q.

Determine if the points (1, 5), (2, 3) and (− 2, − 11) are collinear.

Q. Find the values of x for which the distance between the points P(x, 4) and Q(9, 10) is 10 units.

Q.

If ABC is a triangle right angled at B, then the hypotenuse will be AC.

1. True
2. False

Q.

In a triangle $\Delta ABC,AB=12cm.\angle A={45}^{\circ },\angle B={60}^{\circ }$. The area of triangle ABC will be equal to.

1. $108-72\sqrt{3}c{m}^2$

2. $54-72\sqrt{3}c{m}^2$

3. $108-36\sqrt{3}c{m}^2$

4. $72-36\sqrt{3}c{m}^2$

Q.

In $\Delta ABC,AB=12cm,\angle A={45}^{\circ },\angle B={30}^{\circ }.$ Then the area of the triangle ABC is

1. $3(\sqrt{6}+1)c{m}^2$

2. $6(\sqrt{3}+1)c{m}^2$

3. $6(\sqrt{3}-1)c{m}^2$

4. $3(\sqrt{6}-1)c{m}^2$

Q. What is the distance between the points A (c, 0) and B (0, −c)?

Q. A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Q.

Check whether (5, − 2), (6, 4) and (7, − 2) are the vertices of an isosceles triangle.

Q.

In $\Delta ABC,AB=10cm,\angle A={40}^{\circ },\angle B={80}^{\circ }.$ The area of triangle ABC will be equal to

$\left[\begin{array}{c}tan{40}^{\circ }=0.83\\ tan{80}^{\circ }=5.6\end{array}\right]$

1. $50.2c{m}^2$

2. $44.18c{m}^2$

3. $30.16c{m}^2$

4. $36.14c{m}^2$

Q. In $∆$ABC , the coordinates of vertex A are (0, $-$1) and D (1, 0) and E(0, 10) respectively the mid-points of the sides AB and AC . If F is the mid-points of the side BC , find the area of $∆$ DEF.

Q.

In the given triangle $ABC,AB=8cm,\angle A={50}^{\circ },\angle B={70}^{\circ }$ as shown in the figure. The perpendicular distance from C to AB is equal to $[tan{50}^{\circ }=1.2,tan{70}^{\circ }=2.75]$

1. 10.6 cm

2. 8.6 cm

3. 6.69 cm

4. 7.6 cm

Q.

Find the values of y for which the distance between the points P (2, − 3) and Q (10, y) is 10 units.

Q. A vertically straight tree, 15 m high, is broken by the wind in such a way that it top just touches the ground and makes an angle of 60° with the ground. At what height from the ground did the tree break?

Q. The coordinates of the point on X-axis which are equidistant from the points (−3, 4) and (2, 5) are

(a) (20, 0)

(b) (−23, 0)

(c) $\left(\frac{4}{5},0\right)$

(d) None of these

Q.

In $\Delta ABC,\angle A=\angle B={30}^{\circ },AB=12cm$, the area of the triangle will be equal to

1. $24\sqrt{3}c{m}^2$

2. $12\sqrt{3}c{m}^2$

3. $18\sqrt{3}c{m}^2$

4. $6\sqrt{3}c{m}^2$

Q. The distance between the points A(0, 5) and B(−5, 0) is

(a) 5 units
(b) $2\sqrt{5}$ units
(c) $\sqrt{10}$ units
(d) $5\sqrt{2}$ units

Q.

In $\Delta ABC$ given below, AB = 10 cm, AC = 8 cm and $\angle A={45}^{\circ }$. The find the area of $\Delta ABC$.

1. $10\sqrt{2}c{m}^2$

2. $20c{m}^2$

3. $20\sqrt{2}c{m}^2$

4. $5\sqrt{2}c{m}^2$

Q. $\Delta ABC\sim \Delta PQR$ and $\frac{AB}{PQ}=2$. Then $\frac{AC}{PR}=$

1. 2
2. $\frac{1}{2}$
3. 1

Q. AOBC is a rectangle whose three vertices are A(0, 3), O(0, 0) and B(5, 0). Length of each of its diagonal is

(a) 5 units
(b) 3 units
(c) $\sqrt{34}$ units
(d) 4 units

Q. A circle drawn with origin as the centre passes through the point $A\left(\frac{13}{2},0\right)$. Which of the following points does not lie inside the circle?

(a) $\left(\frac{-3}{4},1\right)$
(b) $\left(2,\frac{7}{3}\right)$
(c) $\left(5,\frac{-1}{3}\right)$
(d) $\left(-6,\frac{5}{2}\right)$

Q.

Determine whether the two figures are similar. Explain your reasoning.

Q. If the distance between the points (4, p) and (1, 0) is 5, then p =

(a) ± 4

(b) 4

(c) −4

(d) 0

Q. In a $\Delta ABC$, line CD is perpendicular to AB. $\angle CAD={45}^{\circ }$ and AB = 20 cm. The area of triangle ABC is $40c{m}^2$. The length of AC will be equal to

1. 4 cm
2. $4\sqrt{2}cm$
3. 8 cm
4. $8\sqrt{2}cm$

Q.

In $\Delta ABC$, $\angle A={60}^{\circ },\angle B={45}^{\circ }andAB=10cm$ as shown in the figure. Then the area of the triangle is equal to

1. $25(3+\sqrt{3})c{m}^2$

2. $50(3-\sqrt{3})c{m}^2$

3. $25(3-\sqrt{3})c{m}^2$

4. $50(3+\sqrt{3})c{m}^2$

Q. $\Delta ABC\sim \Delta PQR$ and $\frac{AB}{PQ}=2$. Then $\frac{AC}{PR}=$

1. 2
2. $\frac{1}{2}$
3. 1

Q.

In triangle $ABC,\angle A={60}^{\circ },\angle B={45}^{\circ }$ and AB=20cm. Then the area of the triangle is equal to

1. $300-100\sqrt{3}c{m}^2$

2. $300+100\sqrt{3}c{m}^2$

3. $300\sqrt{3}+100c{m}^2$

4. $300\sqrt{3}-100c{m}^2$