A right triangle has a hypotenuse equal to 10 and an altitude to the hypotenuse equal to 6. Find the area of the triangle.





Before we begin solving the problem, let's first understand the given information. We are given a right triangle, which means one of the angles is 90 degrees. We are also given the length of the hypotenuse, which is 10, and the length of the altitude to the hypotenuse, which is 6. We need to find the area of the triangle.



The altitude of a right triangle is a line segment drawn from the vertex of the right angle to the hypotenuse, which is perpendicular to the hypotenuse. We can use the Altitude Formula to find the length of the other two sides of the triangle.

Let's assume that the legs of the triangle are a and b, and the hypotenuse is c. The Altitude Formula states that:

altitude^2 = ab

In our case, the altitude is 6, and the hypotenuse is 10. So we can write:

6^2 = ab

Simplifying, we get:

36 = ab



We can also use the Pythagorean Theorem to find the length of the legs of the triangle. The Pythagorean Theorem states that:

a^2 + b^2 = c^2

In our case, c is 10. So we can write:

a^2 + b^2 = 10^2

Simplifying, we get:

a^2 + b^2 = 100

We know that ab is 36 from Step 2. We can use this information to solve for a and b. Solving for b, we get:

b = 36/a

Substituting this into the Pythagorean Theorem equation, we get:

a^2 + (36/a)^2 = 100

Multiplying both sides by a^2, we get:

a^4 + 1296 = 100a^2

Rearranging, we get:

a^4 - 100a^2 + 1296 = 0

This is a quadratic equation in terms of a^2. We can solve for a^2 using the quadratic formula:

a^2 = [100 ± sqrt(100^2 - 4*1*1296)]/2

a^2 = [100 ± sqrt(10000 - 5184)]/2

a^2 = [100 ± sqrt(4816)]/2

a^2 ≈ 26.78 or a^2 ≈ 73.22

Since a^2 + b^2 = 100, we can use the value of a^2 to find b^2:

b^2 = 100 - a^2

b^2 ≈ 73.22 or b^2 ≈ 26.78

Since we know that a and b are the lengths of the legs of the triangle, and we know that the hypotenuse is 10, we can determine which

Using pythagoras therom we can find out base n here base =8n height = 6 ( given)Now ar of trngl = 1/2 × b × h==》 1/2 × 8 × 6 = 24sq unit