QQuestion
In the given diagram, △ADB and △ACB are two right-angled triangles with ∠ADB = ∠BCA = 90°. If AB = 10 cm, AD = 6 cm, BC = 2.4 cm and DP = 4.5 cm:
(a) Prove that △APD ∼ △BPC.
(b) Find the lengths of BD and PB.
(c) Hence, find the length of PA.
(d) Find area of △APD : area of △BPC.
✓Answer
(a) In △APD and △BPC:
∠APD = ∠BPC because they are vertically opposite angles.
∠ADP = ∠BCP = 90°.
Therefore, △APD ∼ △BPC by the AA criterion.
(b) In right-angled △ADB, by the Pythagoras theorem:
AB² = AD² + BD²
⇒ 10² = 6² + BD²
⇒ 100 = 36 + BD²
⇒ BD² = 64
⇒ BD = √64
⇒ BD = 8 cm
Since BD = BP + PD:
PB = BD − PD
⇒ PB = 8 − 4.5
⇒ PB = 3.5 cm
(c) In right-angled △APD:
AP² = AD² + DP²
⇒ AP² = 6² + 4.5²
⇒ AP² = 36 + 20.25
⇒ AP² = 56.25
⇒ AP = √56.25
⇒ AP = 7.5 cm
(d) The ratio of the areas of similar triangles equals the square of the ratio of corresponding sides.
\frac{\text{Area of }\triangle APD}{\text{Area of }\triangle BPC}=\frac{AD^2}{BC^2}⇒ Area of △APD : Area of △BPC = 6² : 2.4²
⇒ 36 : 5.76
Multiplying both terms by 100:
⇒ 3600 : 576
Dividing by 144:
⇒ 25 : 4
Therefore, area of △APD : area of △BPC = 25 : 4.
Related Questions
More ICSE Hard level questions
In △ABC, ∠ABC = 90°, AB = 20 cm and AC = 25 cm. DE is perpendicular to AC such that ∠DEA = 90° and DE = 3 cm, as shown. (a) Prove that △ABC ∼ △AED. (b) Find BC, AD and AE. (c) If BCED represents land on a map whose actual area is 576 m², find the scale factor of the map.
In the given diagram, △ABC ∼ △PQR. If AD and PS are bisectors of ∠BAC and ∠QPR respectively, then: (a) △ABC ∼ △PQS (b) △ABD ∼ △PQS (c) △ABD ∼ △PSR (d) △ABC ∼ △PSR
In the given figure, AC ∥ DE ∥ BF. If AC = 24 cm, EG = 8 cm, GB = 16 cm and BF = 30 cm: (a) Prove △GED ∼ △GBF (b) Find DE (c) Find DB : AB.
In the given figure, ∠BAP = ∠DCP = 70°, PC = 6 cm and CA = 4 cm. Then PD : DB is: (a) 5 : 3 (b) 3 : 5 (c) 3 : 2 (d) 2 : 3
In the given diagram, △ABC ∼ △EFG. If ∠ABC = ∠EFG = 60°, then the length of side FG is: (a) 15 cm (b) 20 cm (c) 25 cm (d) 30 cm
In trapezium ABCD, BC ∥ AD and AD = 4 cm. Diagonals AC and BD meet at O such that AO/OC = DO/OB = 1/2. Find the length of BC.
Perimeters of two similar triangles are 27 cm and 16 cm. If the length of a side of the first triangle is 9 cm, then find the length of the corresponding side of the second triangle
If the lengths of three sides of two triangles are in proportion, then which type of triangle is this?
In ΔABC, L and M are two points on the sides AC and BC respectively such that LM || AB and AL are (x - 2) units, AC = 2x + 3 units, BM = (x – 3) units and BC = 2x units. Determine the value of x.
DE ∥ BC of ΔABC where D and E are two points on AB and AC, respectively. If AD = 5 cm, DB = 6 cm, and AE = 7.5 cm, calculate the length of AC.
Two triangles are similar if their corresponding sides are ___.
Use ruler and compass to answer this question. Construct ∠ABC = 90°, where AB = 6 cm and BC = 8 cm. (a) Construct the locus of points equidistant from B and C. (b) Construct the locus of points equidistant from A and B. (c) Mark the point which satisfies both conditions (a) and (b) as O. Construct the locus of points keeping a fixed distance OA from the fixed point O. (d) Construct the locus of points which are equidistant from BA and BC.