The segments formed by the altitude to the hypotenuse of a right triangle are 8 and 10. Find the shorter leg of that triangle.
Help please! I am confused on which formula to use or if I should use the Pythagorean Theorem.
Make a sketch of a right-angled triangle ABC with angleB = 90°
Draw in the altitude from C to hypotenuse AC to meet it at D.
You now have 3 similar right-angled triangles. (easy to see by considering the angles)
I will list them, with corresponding vertices in the same column
ABC
ADB
BDC
Using the last two:
AD/DB = BD/DC
BD^2 = ADxDC
BD^2 = 10(8) = 80
Now in the smaller triangle
BD^2 + DC^2 = BC^2
80 + 64 = BC^2
BC = √144 = 12
If needed, you can find any of the remaining sides and any of the angles.
Thank you so much!!
welcome!
The segments formed by the altitude
to the hypotenuse of a right triangle
are 8 and 10. Find the shorter leg
of that triangle
To find the shorter leg of a right triangle, you can use either the Pythagorean Theorem or the properties of similar triangles.
Let's first try using the Pythagorean Theorem. The Pythagorean Theorem states that 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, let's assume that the shorter leg of the triangle is x. The longer leg can be represented as (8 + 10 - x) because the longer leg is the sum of the segments formed by the altitude.
So, according to the Pythagorean Theorem, we have:
x^2 + [(8 + 10 - x)^2] = (hypotenuse)^2
Simplifying this equation will give us a quadratic equation. Then you can solve for x by using your preferred method, such as factoring or using the quadratic formula.
Alternatively, you can use the concept of similar triangles. In a right triangle, the altitude to the hypotenuse divides the triangle into two smaller similar triangles. The ratio of the sides in these similar triangles is equal to the ratio of their corresponding altitudes.
In this case, the altitude divides the hypotenuse into two segments of length 8 and 10. So, the ratio of the lengths of the sides of the smaller triangles can be written as x/(8 + 10 - x) = 8/10.
By cross multiplying this proportion, you can obtain a quadratic equation which can be solved to find the value of x.
Both methods will lead you to find the value of the shorter leg.