QUESTION 1
Let the third side of the right angle triangle with sides
be
.
Then, from the Pythagoras Theorem;
![l^2=x^2+6^2](https://tex.z-dn.net/?f=l%5E2%3Dx%5E2%2B6%5E2)
![l^2=x^2+36](https://tex.z-dn.net/?f=l%5E2%3Dx%5E2%2B36)
Let the hypotenuse of the right angle triangle with sides 2,6 be
.
Then;
![m^2=6^2+2^2](https://tex.z-dn.net/?f=m%5E2%3D6%5E2%2B2%5E2)
![m^2=36+4](https://tex.z-dn.net/?f=m%5E2%3D36%2B4)
![m^2=40](https://tex.z-dn.net/?f=m%5E2%3D40)
Using the bigger right angle triangle,
![(x+2)^2=m^2+l^2](https://tex.z-dn.net/?f=%28x%2B2%29%5E2%3Dm%5E2%2Bl%5E2)
![\Rightarrow (x+2)^2=40+x^2+36](https://tex.z-dn.net/?f=%5CRightarrow%20%28x%2B2%29%5E2%3D40%2Bx%5E2%2B36)
![\Rightarrow x^2+2x+4=40+x^2+36](https://tex.z-dn.net/?f=%5CRightarrow%20x%5E2%2B2x%2B4%3D40%2Bx%5E2%2B36)
Group similar terms;
![x^2-x^2+2x=40+36-4](https://tex.z-dn.net/?f=x%5E2-x%5E2%2B2x%3D40%2B36-4)
![\Rightarrow 2x=72](https://tex.z-dn.net/?f=%5CRightarrow%202x%3D72)
![\Rightarrow x=36](https://tex.z-dn.net/?f=%5CRightarrow%20x%3D36)
QUESTION 2
Let the hypotenuse of the triangle with sides (x+2),4 be
.
Then, ![k^2=(x+2)^2+4^2](https://tex.z-dn.net/?f=k%5E2%3D%28x%2B2%29%5E2%2B4%5E2)
![\Rightarrow k^2=(x+2)^2+16](https://tex.z-dn.net/?f=%5CRightarrow%20k%5E2%3D%28x%2B2%29%5E2%2B16)
Let the hypotenuse of the right triangle with sides 2,4 be
.
Then; we have ![t^2=2^2+4^2](https://tex.z-dn.net/?f=t%5E2%3D2%5E2%2B4%5E2)
![t^2=4+16](https://tex.z-dn.net/?f=t%5E2%3D4%2B16)
![t^2=20](https://tex.z-dn.net/?f=t%5E2%3D20)
We apply the Pythagoras Theorem to the bigger right angle triangle to obtain;
![[(x+2)+2]^2=k^2+t^2](https://tex.z-dn.net/?f=%5B%28x%2B2%29%2B2%5D%5E2%3Dk%5E2%2Bt%5E2)
![(x+4)^2=(x+2)^2+16+20](https://tex.z-dn.net/?f=%28x%2B4%29%5E2%3D%28x%2B2%29%5E2%2B16%2B20)
![x^2+8x+16=x^2+4x+4+16+20](https://tex.z-dn.net/?f=x%5E2%2B8x%2B16%3Dx%5E2%2B4x%2B4%2B16%2B20)
![x^2-x^2+8x-4x=4+16+20-16](https://tex.z-dn.net/?f=x%5E2-x%5E2%2B8x-4x%3D4%2B16%2B20-16)
![4x=24](https://tex.z-dn.net/?f=4x%3D24)
![x=6](https://tex.z-dn.net/?f=x%3D6)
QUESTION 3
Let the hypotenuse of the triangle with sides (x+8),10 be
.
Then, ![p^2=(x+8)^2+10^2](https://tex.z-dn.net/?f=p%5E2%3D%28x%2B8%29%5E2%2B10%5E2)
![\Rightarrow p^2=(x+8)^2+100](https://tex.z-dn.net/?f=%5CRightarrow%20p%5E2%3D%28x%2B8%29%5E2%2B100)
Let the hypotenuse of the right triangle with sides 5,10 be
.
Then; we have ![q^2=5^2+10^2](https://tex.z-dn.net/?f=q%5E2%3D5%5E2%2B10%5E2)
![q^2=25+100](https://tex.z-dn.net/?f=q%5E2%3D25%2B100)
![q^2=125](https://tex.z-dn.net/?f=q%5E2%3D125)
We apply the Pythagoras Theorem to the bigger right angle triangle to obtain;
![[(x+8)+5]^2=p^2+q^2](https://tex.z-dn.net/?f=%5B%28x%2B8%29%2B5%5D%5E2%3Dp%5E2%2Bq%5E2)
![(x+13)^2=(x+8)^2+100+125](https://tex.z-dn.net/?f=%28x%2B13%29%5E2%3D%28x%2B8%29%5E2%2B100%2B125)
![x^2+26x+169=x^2+16x+64+225](https://tex.z-dn.net/?f=x%5E2%2B26x%2B169%3Dx%5E2%2B16x%2B64%2B225)
![x^2-x^2+26x-16x=64+225-169](https://tex.z-dn.net/?f=x%5E2-x%5E2%2B26x-16x%3D64%2B225-169)
![10x=120](https://tex.z-dn.net/?f=10x%3D120)
![x=12](https://tex.z-dn.net/?f=x%3D12)
QUESTION 4
Let the height of the triangle be H;
Then ![H^2+4^2=8^2](https://tex.z-dn.net/?f=H%5E2%2B4%5E2%3D8%5E2)
![H^2=8^2-4^2](https://tex.z-dn.net/?f=H%5E2%3D8%5E2-4%5E2)
![H^2=64-16](https://tex.z-dn.net/?f=H%5E2%3D64-16)
![H^2=48](https://tex.z-dn.net/?f=H%5E2%3D48)
Let the hypotenuse of the triangle with sides H,x be r.
Then;
![r^2=H^2+x^2](https://tex.z-dn.net/?f=r%5E2%3DH%5E2%2Bx%5E2)
This implies that;
![r^2=48+x^2](https://tex.z-dn.net/?f=r%5E2%3D48%2Bx%5E2)
We apply Pythagoras Theorem to the bigger triangle to get;
![(4+x)^2=8^2+r^2](https://tex.z-dn.net/?f=%284%2Bx%29%5E2%3D8%5E2%2Br%5E2)
This implies that;
![(4+x)^2=8^2+x^2+48](https://tex.z-dn.net/?f=%284%2Bx%29%5E2%3D8%5E2%2Bx%5E2%2B48)
![x^2+8x+16=64+x^2+48](https://tex.z-dn.net/?f=x%5E2%2B8x%2B16%3D64%2Bx%5E2%2B48)
![x^2-x^2+8x=64+48-16](https://tex.z-dn.net/?f=x%5E2-x%5E2%2B8x%3D64%2B48-16)
![8x=96](https://tex.z-dn.net/?f=8x%3D96)
![x=12](https://tex.z-dn.net/?f=x%3D12)
QUESTION 5
Let the height of this triangle be c.
Then; ![c^2+9^2=12^2](https://tex.z-dn.net/?f=c%5E2%2B9%5E2%3D12%5E2)
![c^2+81=144](https://tex.z-dn.net/?f=c%5E2%2B81%3D144)
![c^2=144-81](https://tex.z-dn.net/?f=c%5E2%3D144-81)
![c^2=63](https://tex.z-dn.net/?f=c%5E2%3D63)
Let the hypotenuse of the right triangle with sides x,c be j.
Then;
![j^2=c^2+x^2](https://tex.z-dn.net/?f=j%5E2%3Dc%5E2%2Bx%5E2)
![j^2=63+x^2](https://tex.z-dn.net/?f=j%5E2%3D63%2Bx%5E2)
We apply Pythagoras Theorem to the bigger right triangle to obtain;
![(x+9)^2=j^2+12^2](https://tex.z-dn.net/?f=%28x%2B9%29%5E2%3Dj%5E2%2B12%5E2)
![(x+9)^2=63+x^2+12^2](https://tex.z-dn.net/?f=%28x%2B9%29%5E2%3D63%2Bx%5E2%2B12%5E2)
![x^2+18x+81=63+x^2+144](https://tex.z-dn.net/?f=x%5E2%2B18x%2B81%3D63%2Bx%5E2%2B144)
![x^2-x^2+18x=63+144-81](https://tex.z-dn.net/?f=x%5E2-x%5E2%2B18x%3D63%2B144-81)
![18x=126](https://tex.z-dn.net/?f=18x%3D126)
![x=7](https://tex.z-dn.net/?f=x%3D7)
QUESTION 6
Let the height be g.
Then;
![g^2+3^2=x^2](https://tex.z-dn.net/?f=g%5E2%2B3%5E2%3Dx%5E2)
![g^2=x^2-9](https://tex.z-dn.net/?f=g%5E2%3Dx%5E2-9)
Let the hypotenuse of the triangle with sides g,24, be b.
Then
![b^2=24^2+g^2](https://tex.z-dn.net/?f=b%5E2%3D24%5E2%2Bg%5E2)
![b^2=24^2+x^2-9](https://tex.z-dn.net/?f=b%5E2%3D24%5E2%2Bx%5E2-9)
![b^2=576+x^2-9](https://tex.z-dn.net/?f=b%5E2%3D576%2Bx%5E2-9)
![b^2=x^2+567](https://tex.z-dn.net/?f=b%5E2%3Dx%5E2%2B567)
We apply Pythagaoras Theorem to the bigger right triangle to get;
![x^2+b^2=27^2](https://tex.z-dn.net/?f=x%5E2%2Bb%5E2%3D27%5E2)
This implies that;
![x^2+x^2+567=27^2](https://tex.z-dn.net/?f=x%5E2%2Bx%5E2%2B567%3D27%5E2)
![x^2+x^2+567=729](https://tex.z-dn.net/?f=x%5E2%2Bx%5E2%2B567%3D729)
![x^2+x^2=729-567](https://tex.z-dn.net/?f=x%5E2%2Bx%5E2%3D729-567)
![2x^2=162](https://tex.z-dn.net/?f=2x%5E2%3D162)
![x^2=81](https://tex.z-dn.net/?f=x%5E2%3D81)
Take the positive square root of both sides.
![x=\sqrt{81}](https://tex.z-dn.net/?f=x%3D%5Csqrt%7B81%7D)
![x=9](https://tex.z-dn.net/?f=x%3D9)
QUESTION 7
Let the hypotenuse of the smaller right triangle be; n.
Then;
![n^2=x^2+2^2](https://tex.z-dn.net/?f=n%5E2%3Dx%5E2%2B2%5E2)
![n^2=x^2+4](https://tex.z-dn.net/?f=n%5E2%3Dx%5E2%2B4)
Let f be the hypotenuse of the right triangle with sides 2,(x+3), be f.
Then;
![f^2=2^2+(x+3)^2](https://tex.z-dn.net/?f=f%5E2%3D2%5E2%2B%28x%2B3%29%5E2)
![f^2=4+(x+3)^2](https://tex.z-dn.net/?f=f%5E2%3D4%2B%28x%2B3%29%5E2)
We apply Pythagoras Theorem to the bigger right triangle to get;
![(2x+3)^2=f^2+n^2](https://tex.z-dn.net/?f=%282x%2B3%29%5E2%3Df%5E2%2Bn%5E2)
![(2x+3)^2=4+(x+3)^2+x^2+4](https://tex.z-dn.net/?f=%282x%2B3%29%5E2%3D4%2B%28x%2B3%29%5E2%2Bx%5E2%2B4)
![4x^2+12x+9=4+x^2+6x+9+x^2+4](https://tex.z-dn.net/?f=4x%5E2%2B12x%2B9%3D4%2Bx%5E2%2B6x%2B9%2Bx%5E2%2B4)
![4x^2-2x^2+12x-6x=4+9+4-9](https://tex.z-dn.net/?f=4x%5E2-2x%5E2%2B12x-6x%3D4%2B9%2B4-9)
![2x^2+6x-8=0](https://tex.z-dn.net/?f=2x%5E2%2B6x-8%3D0)
![x^2+3x-4=0](https://tex.z-dn.net/?f=x%5E2%2B3x-4%3D0)
![(x-1)(x+4)=0](https://tex.z-dn.net/?f=%28x-1%29%28x%2B4%29%3D0)
![x=1,x=-4](https://tex.z-dn.net/?f=x%3D1%2Cx%3D-4)
We are dealing with length.
![\therefore x=1](https://tex.z-dn.net/?f=%5Ctherefore%20x%3D1)
QUESTION 8.
We apply the leg theorem to obtain;
![x(x+5)=6^2](https://tex.z-dn.net/?f=x%28x%2B5%29%3D6%5E2)
![x^2+5x=36](https://tex.z-dn.net/?f=x%5E2%2B5x%3D36)
![x^2+5x-36=0](https://tex.z-dn.net/?f=x%5E2%2B5x-36%3D0)
![(x+9)(x-4)=0](https://tex.z-dn.net/?f=%28x%2B9%29%28x-4%29%3D0)
![x=-9,x=4](https://tex.z-dn.net/?f=x%3D-9%2Cx%3D4)
We discard the negative value;
![\therefore x=4](https://tex.z-dn.net/?f=%5Ctherefore%20x%3D4)
QUESTION 9;
We apply the leg theorem again;
![10^2=x(x+15)](https://tex.z-dn.net/?f=10%5E2%3Dx%28x%2B15%29)
![100=x^2+15x](https://tex.z-dn.net/?f=100%3Dx%5E2%2B15x)
![x^2+15x-100=0](https://tex.z-dn.net/?f=x%5E2%2B15x-100%3D0)
Factor;
![(x-5)(x+20)=0](https://tex.z-dn.net/?f=%28x-5%29%28x%2B20%29%3D0)
![x=5,x=-20](https://tex.z-dn.net/?f=x%3D5%2Cx%3D-20)
Discard the negative value;
![x=5](https://tex.z-dn.net/?f=x%3D5)
QUESTION 10
According to the leg theorem;
The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the portion of the hypotenuse adjacent to that leg.
We apply the leg theorem to get;
![8^2=16x](https://tex.z-dn.net/?f=8%5E2%3D16x)
![64=16x](https://tex.z-dn.net/?f=64%3D16x)
units.
QUESTION 11
See attachment
Question 12
See attachment