Advertisements
Advertisements
प्रश्न
In the given figure, ΔPQR is a right-angled triangle with ∠PQR = 90°. QS is perpendicular to PR. Prove that pq = rx.
उत्तर
Given: ∠PQR = 90° and QS ⊥ PR.
So, ∠QSR = ∠QSP = 90°
Now, in ΔPQR and ΔQSR,
∠QRP ≅ ∠SRQ ......[Common angle]
∠PQR ≅ ∠QSR ......[Each angle is equal to 90°]
So, according to the AA similarity criterion,
ΔPQR ∼ ΔQSR
∴ `(PR)/(QR) = (PQ)/(QS)` .....[C.S.S.T.]
⇒ `r/q = p/x`
⇒ x × r = p × q
⇒ pq = rx
Hence proved.
APPEARS IN
संबंधित प्रश्न
State which pair of triangles in the given figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:
State which pair of triangles in the following figure are similar. Write the similarity criterion used by you for answering the question, and also write the pairs of similar triangles in the symbolic form:
In the following figure, `("QR")/("QS") = ("QT")/("PR")` and ∠1 = ∠2. Show that ΔPQS ~ ΔTQR.
Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that ΔABC ~ ΔPQR.
If AD and PM are medians of triangles ABC and PQR, respectively where ΔABC ~ ΔPQR, prove that `("AB")/("PQ") = ("AD")/("PM")`.
The sides of certain triangles are given below. Determine which of them right triangles are.
9cm, 16cm, 18cm
The sides of certain triangles are given below. Determine which of them right triangles are.
7cm, 24cm, 25cm
The sides of certain triangles are given below. Determine which of them right triangles are.
1.4cm, 4.8cm, 5cm
In ΔABC~ΔDEF such that 2AB = DE and BC = 6cm, find EF.
If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true?
Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of the other triangle. Are the two triangles similar? Why?
In figure, if DE || BC, find the ratio of ar(ADE) and ar(DECB).
A 15 metres high tower casts a shadow 24 metres long at a certain time and at the same time, a telephone pole casts a shadow 16 metres long. Find the height of the telephone pole.
If in two triangles DEF and PQR, ∠D = ∠Q and ∠R = ∠E, then which of the following is not true?
In figure, if ∠D = ∠C, then it is true that ΔADE ~ ΔACB? Why?
In the figure with ΔABC, P, Q, R are the mid-points of AB, AC and BC respectively. Then prove that the four triangles formed are congruent to each other.
A tangent ADB is drawn to a circle at D whose centre is C. Also, PQ is a chord parallel to AB and ∠QDB = 50°. Find the value of ∠PDQ.
The sum of two angles of a triangle is 150°, and their difference is 30°. Find the angles.
In ΔABC, AP ⊥ BC, BQ ⊥ AC. If AP = 7, BQ = 8 and BC = 12, then find AC.
Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3 RS. Find the ratio of the areas of triangles POQ and ROS.
