Question

20 POINTS FOR 3 QUESTIONS!!!!

Given: ∆ABC, AB = BC, m∠1<90°


Perimeter of ∆ABC = 25


Difference between the two sides is 4


Find: AB, BC, AC



List the sides of ΔRST in in ascending order (shortest to longest) if:


m∠R = 3x+42°, m∠S = 4x−11°, and m∠T = x+13°



List the sides of ΔRST in in ascending order (shortest to longest) if:


m∠R = 2x+11°, m∠S = 3x+23°, and m∠T = x+42°

Answer 1

Answer with Step-by-step explanation:

1.In triangle ABC

AB=BC

Let AB=BC=x and AC=y

Perimeter of triangle ABC=25

`x+x+y=25`

`2x+y=25`...(1)

`x-y=4`...(2)

Adding equation (1) and (2)

`3x=29`

`x=(29)/(3)=9.67`

Substitute x=9.67 in equation (2)

`9.67-y=4`

`y=9.67-4=5.67`

`AB=BC=9.67`

`AC=5.67`

2.
`m\angle R=2x+11`

`m\angle S=3x+23`

`m\angle T=x+42`

`m\angle R+m\angle S+m\angle T=180^(\circ)`

By using triangle angle sum property

Substitute the values then we get

`3x+42+4x-11+x+13=180`

`8x+44=180`

`8x=180-44=136`

`x=(136)/(8)=17`

Substitute the value

`m\angle R=3(17)+42=93^(\circ)`

`m\angle S=4(17)-11=57^(\circ)`

`m\angle T=17+13=30^(\circ)`

`m\angle R>m\angle S`

`m\angle S>m\angle T`

`ST>RT` (Side ST is opposite to angle R, Side RT is opposite to angle S

`RT>RS` (side RS is opposite to angle T)

When a>b

Then , opposite side of a> opposite side of b

RS<RT<ST

3.
`m\angle R=2x+11`

`m\angle S=3x+23`

`\angle T=x+42`

`m\angle R+m\angle S+m\angle T=180^(\circ)`

By using triangle angle sum property

Substitute the values then we get

`2x+11+3x+23+x+42=180`

`6x+76=180`

`6x=180-76`

`6x=104`

`x=(104)/(6)=17.3`

Substitute the value

`m\angle R=2(17.3)+11=45.6`

`m\angle S=3(17.3)+23=74.9`

`m\angle T=17.3=42=59.3`

`m\angle S>m\angle T`

`m\angle T>m\angle R`

RT>RS

RS>ST

ST<RS<RT