In right triangle ABC, measure of angle ACB= 90 and segment CD is the altitude to the hypotneuse AB. Ad= 6 cm and BD= 4 cm. Find the length of segment CD to the nearest hundredth.
The two little triangles are similar to the big one because they are all right triangles sharing one angle so sharing all angles (sum is 180)
A = CAD = BCD
B = CBD = ACD
CD/6 = 4/CD similar triangles
CD^2 = 4*6
CD = 2 sqrt 6
To find the length of segment CD, we first need to find the length of the hypotenuse AB.
We can use the Pythagorean theorem to find the length of AB. 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, we have AD = 6 cm and BD = 4 cm. We can use these values to find the length of AB.
According to the Pythagorean theorem, AB^2 = AD^2 + BD^2.
Substituting the given values, AB^2 = 6^2 + 4^2.
Simplifying, AB^2 = 36 + 16.
AB^2 = 52.
To find the length of AB, we take the square root of both sides of the equation:
AB = √52.
Approximating the value of √52 to the nearest hundredth, we get AB ≈ 7.21 cm.
Now that we know the length of AB, we can find the length of CD.
CD is the altitude to the hypotenuse AB, so it divides AB into two segments, and it is perpendicular to AB.
Since CD is perpendicular to AB, triangle BCD is similar to triangle ABC by the Angle-Angle (AA) similarity theorem, which states that two triangles are similar if they have two pairs of corresponding angles that are equal.
In triangle ABC, angle ACB = 90 degrees, and in triangle BCD, angle BDC = 90 degrees.
Additionally, angle BCD is congruent to angle ABC since they are corresponding angles in the two similar triangles.
Therefore, triangles ABC and BCD are similar, and we can set up the following proportion:
AB/BC = BC/CD.
Substituting AB = 7.21 cm and BC = 6 cm into the proportion, we get:
7.21/6 = 6/CD.
Simplifying, 1.20 = 6/CD.
To solve for CD, we can cross-multiply:
CD = 6/1.20.
CD = 5.
Therefore, the length of segment CD is 5 cm.