Geoemtry Homework Questions? Similar Polygons?

Question

Geometry help questions?
1. If 3/12+x = 2/8+y, then find y:x
2. Find the positive value of k for which k:9 = 10:(k+9)
3. The angle bisector of one angle of the triangle trisects the opposite side. In what ratio does it divide the perimeter of the original triangle?
4. Triangle TOP ~ Triangle HAT, TO = 15, AT = 24, HA = 18, and HT = 20. Find PT.

Answer 1

1. If 3/12+x = 2/8+y, then find y:x

1/4 + x = 1/4 + y

Subtract 1/4 from each side.

x = y

Divide each side by x.

1 = y/x

y : x = 1 : 1.

 

(or)

3/(12+x) = 2/(8 + y)

Cross multiplication.

3(8+y) = 2(12 + x)

24 + 3y = 24 + 2x

Subtract 24 from each side.

3y = 2x

Divide each side by 3x.

y/x = 2/3

Therefore y : x = 2 : 3.

Answer 2

2). k : 9 = 10 : (k + 9)

This equation it can be written as k / 9 = 10 / (k + 9)

Cross multiplication.

k(k + 9) = 10(9)

Distribute terms using distributive property:  a( b + c) = ab + ac

k² + 9k = 90

Subtract 90 from each side.

k² + 9k - 90 = 0

Now solve the equation using factor method.

k² + 15k - 6k - 90 = 0

k(k + 15) - 6(k + 15) = 0

(k + 15)(k - 6) = 0

k + 15 = 0 or k - 6 = 0

If k + 15 = 0 then k = -15

If k - 6 = 0 then k = 6

The positive value of k is 6.

Answer 3

• 4).

Triangle TOP and triangle HAT are similar triangles.

Fundamental theorem on similar triangles : If triangle TOP ∼ triangle HAT, then TO/HA = OP/AT = PT/HT.

TO = 15, AT = 24, HA = 18, and HT = 20.

TO/HA = PT/HT

15/18 = PT/20

PT = (15/18) * 20

= (5/6) * 20

= (5 * 10)/3

= 50/3.

Therefore, PT = 50/3.