Solution:
In Δ M NP
P N ║ AB→→(Given)
So, Corresponding angles are equal.
1. ∠MAB=∠MNP
2. ∠MBA= ∠MPN
In Δ MNP and Δ MAB
∠M is common.
1. ∠MAB=∠MNP
2. ∠MBA= ∠MPN
→ Δ MNP ~ Δ MAB→→(Angle - Angle Similarity)
As, when triangles are similar ,their corresponding sides are proportional.
→ [tex]\frac{MA}{MN}=\frac{MB}{MP}[/tex]→→→CPCT
→→[tex]\frac{67.2-32}{67.2}=\frac{x}{81.9}\\\\ \frac{35.2}{67.2}=\frac{x}{81.9}\\\\ x=42.9[/tex] meter
2. In Δ GEH
∠GED=∠HED→→[GIven]
Using the angle bisector theorem, if in a triangle an angle is bisected, the ratio of the sides adjacent to that angle is equal to ratio of the two segments which are equal to those lengths where the angle bisector cuts the third side.
[tex]\frac{EG}{EH}=\frac{GD}{DH}\\\\ \frac{99.2}{112}=\frac{62}{x+2}[/tex]
→112 × 62= 99.2 × (x+2)
→6944=99.2 × (x+2)
→x+2= 70→→Dividing both sides by 99.2
→x= 70-2
→x=68 feet
3. In Δ ABC
DE ║ BC
If a line is passing through two sides of triangle, parallel to third side it divides the other two sides in same ratio.
→ [tex]\frac{11}{121}=\frac{10}{5 x +10}\\\\ \frac{1}{11}=\frac{2}{x+2}\\\\ x+2=22\\\\ x=22-2\\\\ x=20[/tex] cm
4. Area of Right angled triangle = 210 square inches------(1)
Height of right angled triangle= 35 inches
As, Area of Right angled triangle=
[tex]\frac{1}{2} \times B \times H[/tex]-----(2)
Where, B=Base and H= Height
Equating (1) and (2), we get
→ 210 ×2=35 ×B
→ Base=12 inches
Pythagorean theorem states
(Base)² + (Height)² = (Hypotenuse)²
→12² + 35²= (Hypotenuse)²
→144 +1225= (Hypotenuse)²
→→ (Hypotenuse)²=1369
→→Hypotenuse= 37 inches
