Given: △CDE m∠D=90°, DK ⊥ CE CD:DE=3:5, KE=CK+8 Find: CE

Question

Given: △CDE

       m∠D=90°, 

DK

 ⊥ 

CE

 

       CD:DE=3:5, KE=CK+8

    Find: CE

Answer 1

The problem said "CD:DE=3:5." That's a ratio, not their values. The answer of about 5.83 is a ratio also. CD:DE:CE = 3:5:about 5.83. If using the Pythagorean Theorem (which I heard was not made by Pythagoras) was the only step, there wouldn't need to be Point K. If you haven't used all the given information, you probably haven't solved the problem. What you need to know is that the altitude of a right triangle (DK) makes the left triangle, right triangle, and whole triangle similar. CK:DK = 3:5 and DK:KE = 3:5, so CK:KE = 9:25. KE = CK + 8. Here's the algebra where x = CK

x + 8 = 25x/9

9(x + 8) = 25x

9x + 72 = 25x

72 = 16x

x = 4.5

DK = 7.5, KE = 12.5 (which is 4.5 + 8), and CE = 17. CD is the square root of 76.5, and DE is the square root of 212.5 Using the Pythagorean Theorem shows that all of the length work together.

Answer 2

Here is a suggested strategy.

Since CD:DE = 3:5, let CD be represented by 3x and DE by 5x.

Let CK be represented by y and KE by y + 8.

Each leg of the "big" right triangle is the mean proportional between its own projection on the hypotenuse and the entire hypotenuse.

So we can set up equations like this:

y/3x = 3x / (2y+8) Note: 2y+8 is the length of the entire hypotenuse)

(y+8)/5x = 5x/(2y+8)

There are two equations, and two unknowns. Try cross multiplying setting up two quadratic equations in terms of x and y. Use substitution or elimination method to solve for y (by replacing x^2). Looks like it will be messy.

Another option is to let z = hypotenuse, find expressions for CK and KE in terms of z. Then do the same as above--might not be quite as messy.

Give it a shot and let me know if this works for you.

Answer 3

CD=3

DE=5

a^2+b^2=c^2 (rt triangle)

In this case: CD=a; DE=b; CE=c.

So: 9+25=34^.5=5.83