Question:
In triangles ABC and PQR, \(\angle A = \angle Q\) and \(\angle B = \angle R\), then AB : AC is equal to :
In triangles ABC and PQR, \(\angle A = \angle Q\) and \(\angle B = \angle R\), then AB : AC is equal to :
Show Hint
To find side ratios without confusion, write the similarity statement \(\Delta ABC \sim \Delta QRP\) clearly first.
Then, pick the pairs of letters directly:
- \(AB\) (letters 1 and 2) corresponds to \(QR\) (letters 1 and 2).
- \(AC\) (letters 1 and 3) corresponds to \(QP\) (letters 1 and 3).
This gives \(\frac{AB}{AC} = \frac{QR}{QP}\) immediately!
Then, pick the pairs of letters directly:
- \(AB\) (letters 1 and 2) corresponds to \(QR\) (letters 1 and 2).
- \(AC\) (letters 1 and 3) corresponds to \(QP\) (letters 1 and 3).
This gives \(\frac{AB}{AC} = \frac{QR}{QP}\) immediately!
Updated On: Jul 7, 2026
- PQ : PR
- PQ : QR
- QR : QP
- PR : QR
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We are given two triangles, \(\Delta ABC\) and \(\Delta PQR\), with two pairs of equal corresponding angles: \(\angle A = \angle Q\) and \(\angle B = \angle R\). We need to determine the ratio equal to \(AB : AC\).
Step 2: Key Formula or Approach:
1. If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar by the AA (Angle-Angle) similarity criterion.
2. When writing similarity statements, the order of the vertices must match the corresponding equal angles.
3. For similar triangles, the ratio of corresponding sides is equal.
Step 3: Detailed Explanation:
1. Identify the matching vertices based on the given angle equalities:
- Vertex \(A\) corresponds to vertex \(Q\) (since \(\angle A = \angle Q\)).
- Vertex \(B\) corresponds to vertex \(R\) (since \(\angle B = \angle R\)).
- This implies the third vertex \(C\) must correspond to the third vertex \(P\) (since \(\angle C = \angle P\) by the angle sum property of triangles).
2. Write the formal similarity statement matching the corresponding vertices:
\[ \Delta ABC \sim \Delta QRP \]
3. Since the triangles are similar, the ratios of their corresponding sides must be equal:
\[ \frac{AB}{QR} = \frac{BC}{RP} = \frac{AC}{QP} \]
4. We are asked to find the ratio \(AB : AC\), which is \(\frac{AB}{AC}\).
Using the first and third ratios:
\[ \frac{AB}{QR} = \frac{AC}{QP} \]
Rearrange this equation to group \(AB\) and \(AC\) on one side:
\[ \frac{AB}{AC} = \frac{QR}{QP} \]
Therefore, the ratio \(AB : AC\) is equal to \(QR : QP\).
Step 4: Final Answer:
The ratio \(AB : AC\) is equal to \(QR : QP\), which corresponds to option (C).
We are given two triangles, \(\Delta ABC\) and \(\Delta PQR\), with two pairs of equal corresponding angles: \(\angle A = \angle Q\) and \(\angle B = \angle R\). We need to determine the ratio equal to \(AB : AC\).
Step 2: Key Formula or Approach:
1. If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar by the AA (Angle-Angle) similarity criterion.
2. When writing similarity statements, the order of the vertices must match the corresponding equal angles.
3. For similar triangles, the ratio of corresponding sides is equal.
Step 3: Detailed Explanation:
1. Identify the matching vertices based on the given angle equalities:
- Vertex \(A\) corresponds to vertex \(Q\) (since \(\angle A = \angle Q\)).
- Vertex \(B\) corresponds to vertex \(R\) (since \(\angle B = \angle R\)).
- This implies the third vertex \(C\) must correspond to the third vertex \(P\) (since \(\angle C = \angle P\) by the angle sum property of triangles).
2. Write the formal similarity statement matching the corresponding vertices:
\[ \Delta ABC \sim \Delta QRP \]
3. Since the triangles are similar, the ratios of their corresponding sides must be equal:
\[ \frac{AB}{QR} = \frac{BC}{RP} = \frac{AC}{QP} \]
4. We are asked to find the ratio \(AB : AC\), which is \(\frac{AB}{AC}\).
Using the first and third ratios:
\[ \frac{AB}{QR} = \frac{AC}{QP} \]
Rearrange this equation to group \(AB\) and \(AC\) on one side:
\[ \frac{AB}{AC} = \frac{QR}{QP} \]
Therefore, the ratio \(AB : AC\) is equal to \(QR : QP\).
Step 4: Final Answer:
The ratio \(AB : AC\) is equal to \(QR : QP\), which corresponds to option (C).
Was this answer helpful?
0
0
Top CBSE X Mathematics Questions
- Express each number as a product of its prime factors:
- \(140\)
- \(156\)
- \(3825\)
- \(5005\)
- \(7429\)
- Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
- \(26\) and \(91\)
- \(510\) \(\)and \(92\)
- \(336\) and \(54\)
- Find the LCM and HCF of the following integers by applying the prime factorisation method.
- \(12, 15\) and
- \(17, 23\) and
- \(8, 9\) and \(25\)
- Given that HCF \((306, 657) = 9\) , find LCM \((306, 657)\).
- Check whether \(6n\) can end with the digit \(0\) for any natural number \(n\).
View More Questions
Top CBSE X Similarity of Triangles Questions
- If in triangles $ABC$ and $PQR$, $\frac{AB}{QR} = \frac{BC}{PR}$, then they will be similar, when:
- In a $\triangle ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $BD = CE$. If $\angle B = \angle C$, then show that $DE \parallel BC$.
- If $\triangle ABC \sim \triangle DEF$ and $AB = 4$ cm, $DE = 6$ cm, $EF = 9$ cm, and $FD = 12$ cm, find the perimeter of $\triangle ABC$.
- $ABCD$ is a trapezium with $AB \parallel DC$. $AC$ and $BD$ intersect at $E$. If $\triangle AED \sim \triangle BEC$, then prove that $AD = BC$.
- If in triangles $ABC$ and $PQR$, $\frac{AB}{QR} = \frac{BC}{PR}$, then they will be similar, when:
View More Questions
Top CBSE X Questions
- Read the following source carefully and answer the questions that follow:
Print Comes to India
From 1780, James Augustus Hickey began to edit the Bengal Gazette, a weekly maga zine that described itself as ‘a commercial paper open to all, but influenced by none’. So it was private English enterprise, proud of its independence from colonial influence, that began English printing in India. Hickey published a lot of advertisements, including those that related to the import and sale of slaves. But he also published a lot of gossip about the Company’s senior officials in India. Enraged by this, Governor-General War ren Hastings persecuted Hickey, and encouraged the publication of officially sanctioned newspapers that could counter the flow of information that damaged the image of the colonial government. By the close of the eighteenth century, a number of newspapers and journals appeared in print. There were Indians, too, who began to publish Indian news papers. The first to appear was the weekly Bengal Gazette, brought out by Gangadhar Bhattacharya, who was close to Raja Rammohan Roy. - Read the following source carefully and answer the questions that follow:
Loans from Cooperatives
Besides banks, the other major source of cheap credit in rural areas are the cooperative societies (or cooperatives). Members of a cooperative pool their resources for coopera tion in certain areas. There are several types of cooperatives possible such as farmers cooperatives, weavers cooperatives, industrial workers cooperatives, etc. Krishak Coop erative functions in a village not very far away from Sonpur. It has 2300 farmers as members. It accepts deposits from its members. With these deposits as collateral, the Cooperative has obtained a large loan from the bank. These funds are used to provide loans to members. Once these loans are repaid, another round of lending can take place. Krishak Cooperative provides loans for the purchase of agricultural implements, loans for cultivation and agricultural trade, fishery loans, loans for construction of houses and for a variety of other expenses. - More than three million carbon compounds have been discovered in the field of chemistry. The diversity of these compounds is due to the capacity of carbon atoms for bonding with one another as well as with other atoms. Most of the carbon compounds are poor conductors of electricity and have low melting and boiling points.
- Human digestive system is a tube running from mouth to anus. Its main function is to breakdown complex molecules present in the food which cannot be absorbed as such into smaller molecules. These molecules are absorbed across the walls of the tube and the absorbed food reaches each and every cell of the body where it is utilised for obtaining energy.
- A student has three convex lenses A, B, and C of different focal lengths. He wants to observe the images formed by these lenses on a screen by placing a candle flame at different distances as given in the following table:
Case No. Lens Focal Length Object Distance 1 \(A\) 50 cm 25 cm 2 B 20 cm 60 cm 3 C 15 cm 30 cm
View More Questions