Triangles — Karnataka (SSLC) Class 10 Mathematics Solutions (Free)
Free step-by-step Karnataka (SSLC) Class 10 Mathematics solutions for "Triangles" — important questions with detailed answers, download PDF for board exam preparation.
TL;DR: Free step-by-step Karnataka (SSLC) Class 10 Mathematics solutions for "Triangles" — important questions with detailed answers, download PDF for board…
By Syllab.in · Updated
Q1: State and explain the Basic Proportionality Theorem (Thales' Theorem). What does it tell us about similar triangles?
Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle, then it divides the other two sides in the same ratio.
Statement: In triangle ABC, if DE || BC (DE is parallel to BC), where D is on AB and E is on AC, then:
AD/DB = AE/EC
Explanation:
- The parallel line DE creates two smaller triangles: ADE and ABC
- These triangles are similar (AA similarity — angle A is common, and corresponding angles are equal due to parallel lines)
- Similar triangles have propor…
Q2: Two triangles ABC and PQR are similar. If AB = 4 cm, BC = 6 cm, CA = 5 cm, and PQ = 6 cm, find QR and RP.
Given: Triangle ABC ~ Triangle PQR
AB = 4 cm, BC = 6 cm, CA = 5 cm
PQ = 6 cm
Find: QR and RP
Step 1: Since ABC ~ PQR, the corresponding sides are proportional.
Corresponding sides are: AB ↔ PQ, BC ↔ QR, CA ↔ RP
Step 2: Write the ratio of similarity.
AB/PQ = BC/QR = CA/RP = k (ratio of similarity)
Step 3: Find the ratio k.
k = AB/PQ = 4/6 = 2/3
Step 4: Find QR.
BC/QR = 2/3
6/QR = 2/3
QR = 6 × 3/2 = 9 cm
Step 5: Find RP.
CA/RP = 2/3
5/RP = 2/3
RP = 5 × 3/2 = 7.5 cm
Answer: QR = 9 cm, RP = 7…
Q3: In triangle ABC, DE || BC where D is on AB and E is on AC. If AD = 3 cm, DB = 6 cm, and AE = 2 cm, find EC.
Given: Triangle ABC with DE || BC
D is on AB, E is on AC
AD = 3 cm, DB = 6 cm, AE = 2 cm
Find: EC
Step 1: Apply Basic Proportionality Theorem.
Since DE || BC, we have:
AD/DB = AE/EC
Step 2: Substitute the known values.
3/6 = 2/EC
Step 3: Solve for EC.
1/2 = 2/EC
EC = 2 × 2 = 4 cm
Answer: EC = 4 cm
Verification: AD/DB = 3/6 = 1/2 and AE/EC = 2/4 = 1/2 ✓
Q4: Two similar triangles have corresponding sides in the ratio 3:5. If the area of the smaller triangle is 36 cm², find the area of the larger triangle.
Given: Two similar triangles with sides in ratio 3:5
Area of smaller triangle = 36 cm²
Find: Area of larger triangle
Step 1: Recall the relationship between areas of similar triangles.
When two triangles are similar with sides in ratio k:1, their areas are in ratio k²:1.
Step 2: Here the ratio of sides = 3:5
Ratio of areas = (3)²:(5)² = 9:25
Step 3: Let the area of the larger triangle be A.
Area of smaller triangle : Area of larger triangle = 9:25
36:A = 9:25
Step 4: Solve for A.
36/A = 9/2…
Q5: In triangle ABC, a line parallel to BC intersects AB at D and AC at E such that AD:DB = 2:1. If BC = 15 cm, find DE.
Given: DE || BC in triangle ABC
AD:DB = 2:1, BC = 15 cm
Find: DE
Step 1: Express AD and DB in terms of a variable.
Let AD = 2x and DB = x
Then AB = AD + DB = 2x + x = 3x
Step 2: Find the ratio AD:AB.
AD:AB = 2x:3x = 2:3
Step 3: Triangles ADE and ABC are similar (AA similarity).
Since DE || BC:
∠ADE = ∠ABC (corresponding angles)
∠AED = ∠ACB (corresponding angles)
∠A is common
Step 4: Write the ratio of corresponding sides.
Since triangles ADE and ABC are similar:
AD/AB = DE/BC = AE/AC
Step …
Showing 5 of 9 questions — full solutions on the page.