Question
ABC is a triangle and through A, B, C lines are drawn parallel to BC, CA and AB respectively intersecting at P, Q and R. Prove that the perimeter of $\triangle\text{PQR}$ is double the perimeter of $\triangle\text{ABC}.$
✓
Answer
Clearly ABCQ and ARBC are parallelograms. Therefore, BC = AQ and BC = AR ⇒ AQ = AR ⇒ A is the mid-point of QR Similarly B and C are the mid points of PR and PQ respectively.$\therefore\text{AB}=\Big(\frac{1}{2}\Big)\text{PQ},$ $\text{BC}=\Big(\frac{1}{2}\Big)\text{QR},$ $\text{CA}=\Big(\frac{1}{2}\Big)\text{PR}.$
⇒ PQ = 2AB, QR = 2BC and PR = 2CA ⇒ PQ + QR + RP = 2 (AB + BC + CA) ⇒ Perimeter of $\triangle\text{PQR}=2$ $($perimeter of $\triangle\text{ABC})$
⇒ PQ = 2AB, QR = 2BC and PR = 2CA ⇒ PQ + QR + RP = 2 (AB + BC + CA) ⇒ Perimeter of $\triangle\text{PQR}=2$ $($perimeter of $\triangle\text{ABC})$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Simplify the following:
$(2 x-5 y)^3-(2 x+5 y)^3$
→$(2 x-5 y)^3-(2 x+5 y)^3$
In right angled $\triangle P Q R, \angle Q=90^{\circ}, \angle R=\theta$ and if $\sin \theta=\frac{5}{13}$, then find $\cos \theta$ and $\tan \theta$,
→Find the area of the of the blades of the magnetic compass shown in Fig. below$\big($ Take $\sqrt{11}=3.32\big).$
→Plot the points A(2, 5), B(-2, 2) and C(4, 2) on a graph paper. Join AB, BC and AC. Calculate the area of $\triangle\text{ABC}.$
→Prove that $(a + b + c)^3 - a^3 - b^3 - c^3 = 3(a + b)(b + c)(c + a).$
→AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig). Show that the line PQ is perpendicular bisector of AB.
In the adjoining figure, ABCD is a square and $\triangle\text{EDC}$ is an equilateral triangle. Prove that:
→- AE = BE,
- $\angle\text{DAE}=15^{\circ}.$
If $\text{x}=-\frac{1}{2}$ is zero of the polynomial $p(x) = 8x^3 - ax^2 - x + 2$, Find the value of a.
→Define a trial.
In the given figure, $AB\ ||\ CD$. Find the value of $x$.
→