how long are the congruent legs of isosceles triangle if its base and altitude to the base are 12cm and 8 cm respectively?
To find the length of the congruent legs of an isosceles triangle, we can apply the Pythagorean theorem. In this case, the triangle has a base of 12 cm and an altitude (which is perpendicular to the base) of 8 cm.
Let's label the congruent legs as a and b. Since the triangle is isosceles, we know that a = b.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, the altitude is the hypotenuse, and the lengths of the congruent legs are the other two sides. So we have:
a^2 + b^2 = altitude^2
Since a = b, we can rewrite the equation as:
2a^2 = altitude^2
Substituting the given values, we get:
2a^2 = 8^2
Simplifying further:
2a^2 = 64
Dividing both sides by 2:
a^2 = 32
Now, taking the square root of both sides:
a = √32
Simplifying the square root:
a ≈ 5.66 cm
Therefore, the length of the congruent legs of the isosceles triangle is approximately 5.66 cm.