# AMBQID 64:

In the given figure, the triangle ABC is a right isosceles triangle and BD = BC. Find the value of θ.

Options:

A. 22.5 Degree

A. 30 Degree

A. 7.5 Degree

D. 15 Degree

**Solution:**

To solve the above question, we need to use the following math concepts.

**Concept 1:** The perpendicular distance between the parallel lines is same at any points

**Concept 2:** In a right triangle, the Pythagoras theorem is used. It the square of the hypotenuse is the sum of the squares of the perpendicular and the square of the base.

**Concept 3:** In a right isosceles triangle, the two perpendicular lines are equal and each acute angle is 45 degree.

**Concept 4:** In an isosceles triangle, the perpendicular drawn from the vertex to opposite side bisects the side.

**Concept 5:** In trigonometry, sin(Angle) = Perpendicular / Hypotenuse.

Let the sides of right isosceles triangle AB = AC = a. Then, BC = by using **Concept 2**.

Since AB = AC = a, then angle ABC = 45 degree using concept 3.

Given that BD = BC. Thus BD = .

Now, in isosceles triangle ABC, drawn a perpendicular AM from the vertex A to the opposite side BC. Thus, BM = MC = BC/2 by using **concept 4**. Hence, BM = .

In right triangle ABM, AM = using **concept 2**.

Since the lines AD and MN are parallel, then AM = DN by using **concept 1**. Thus, DN = . Now, let angle DBN = x.

Using **concept 5**, we get sin x = DN / BD = 1/2. Thus, x = 30 degree.

Therefore, θ = 45 degree – 30 degree = 15 degree. The **correct option is D 15 degree**.

**Some Important Questions**

**The figure below is an equilateral triangle with sides of length 6. What is the area of the triangle? **

Since the triangle is an equilateral triangle, its angles are 60°.

Therefore, the vertical line *AX* bisects the angle *BAC*, so it splits it into two congruent angles of 30° each.

So, the two smaller triangles are 30-60-90 triangles. Thus, the ratio of the lengths of the sides is 1: :2.

The area of a triangle is given by the formula , where *b* is the length of the base and *h* is the height. In the figure, the base is 3 + 3 = 6 and the height is 3.

Substitute these values into the formula and calculate the area as A = (1/2) x 6 x 3 = 9

**Which of the following could be the side lengths of a right triangle?**

Options:

A. 4, 5, and 6

A. 4, 9, and 10

A. 5, 10, and 15

D. 5, 12, and 13

If a triangle is a right triangle, then the lengths of its sides satisfy the Pythagorean Theorem,

In a right triangle, the Pythagoras theorem is used. It the square of the hypotenuse is the sum of the squares of the perpendicular and the square of the base. *a ^{2}* +

*b*

^{2}=

*c*.

^{2}To determine which choice is correct, test each set of values by substituting them into the Pythagorean Theorem. Start with the first set of numbers: 3, 13, and 14.

Since the result is not a true equality, the first set of values does not represent the side lengths of a right triangle. Test the other four choices. The only values that satisfy the Pythagorean Theorem are 5, 12, and 13

**The circumference of a circle is 30. What is its area?**

Options:

A. 225

B. 900

C. 400

D. 3000

The circumference of a circle is given by the formula *C* = 2, where *r* is the radius of the circle.

Substitute the given circumference into this formula and solve for *r*.

30 = 2

r = 15

Therefore, the radius of the circle is 15.

Thus, the area of a circle is given by the formula *A*= ^{2}. Substitute the length of the radius into this formula and calculate the area.

*A* =(15)^{2} = 225