17. Side of an equilateral triangle is 4cm. Its area is? 18. Area of a triangle having base 6cm and altitude 8cm is? 19. Assertion (A): The area of an isosceles triangle each of wh... 17. Side of an equilateral triangle is 4cm. Its area is? 18. Area of a triangle having base 6cm and altitude 8cm is? 19. Assertion (A): The area of an isosceles triangle each of whose equal side is 13cm and whose base is 24cm is 60cm². Reason(R): The area of an isosceles triangle having base a and each equal side b is $\frac{b}{4}\sqrt{4a^2 - b^2}$. Are both Assertion (A) and Reason (R) true and Reason(R) is the correct explanation for Assertion (A)? Read more

Steps to Solve

1. Calculate the area of the equilateral triangle

The formula for the area of an equilateral triangle with side $s$ is given by: $$Area = \frac{s^2\sqrt{3}}{4}$$ In this case, $s = 4$ cm. Substitute this value into the formula:

$$Area = \frac{4^2\sqrt{3}}{4} = \frac{16\sqrt{3}}{4} = 4\sqrt{3} \text{ cm}^2$$

2. Calculate the area of the triangle

The area of a triangle is given by the formula:

$$Area = \frac{1}{2} \times base \times altitude$$ Given base = 6 cm and altitude = 8 cm. Substitute these values into the formula:

$$Area = \frac{1}{2} \times 6 \times 8 = 3 \times 8 = 24 \text{ cm}^2$$

3. Verify the area of the isosceles triangle in Assertion (A)

The isosceles triangle has equal sides of 13cm and a base of 24cm. To find the area, we first need to find the height. We can divide the isosceles triangle into two right triangles by drawing an altitude from the vertex where the two equal sides meet, to the midpoint of the base. This altitude represents the height ($h$) of the isosceles triangle. The base of each right triangle is half the base of the isosceles triangle, i.e., $24/2 = 12$ cm. The hypotenuse of each right triangle is 13 cm (the equal side of the isosceles triangle). Now we can use the Pythagorean theorem to find the height ($h$):

$$h^2 + 12^2 = 13^2$$ $$h^2 + 144 = 169$$ $$h^2 = 169 - 144 = 25$$ $$h = \sqrt{25} = 5 \text{ cm}$$ Now, we can calculate the area of the isosceles triangle:

$$Area = \frac{1}{2} \times base \times height$$ $$Area = \frac{1}{2} \times 24 \times 5 = 12 \times 5 = 60 \text{ cm}^2$$ So, Assertion (A) is true.

4. Verify the formula in Reason (R)

The formula for the area of an isosceles triangle with base $a$ and equal sides $b$ is given as: $$Area = \frac{b}{4}\sqrt{4a^2 - b^2}$$ However, according to standard mathematical formulas, the correct formula should be $Area = \frac{a}{4} \sqrt{4b^2 - a^2}$. So Reason (R) gives the wrong formula. So, Reason (R) is false.