In the adjoining figure D is a point on BC such that, ∠ABD = ∠CAD. If AB = 5cm, AD = 4cm and AC = 3cm. Find A (∆ACD) : A (∆BCA).
Last updated date: 19th Mar 2023
•
Total views: 303.9k
•
Views today: 2.83k
Answer
303.9k+ views
Hint: Visualizing the given data in the question draws an appropriate figure. According to the properties of triangles, observe similarities between the two triangles in order to establish a relation between their angles or sides.
Complete step-by-step answer:
Given data,
∠ABD = ∠CAD. AB = 5cm, AD = 4cm and AC = 3cm.
Let us compare ∆ABC and ∆CAD
∠ABD = ∠CAD (given)
∠ACB = ∠ACD (common angle, from the figure)
The Third Angle theorem states, if two angles of a triangle are congruent to two angles in another triangle, then the third pair of angles are also congruent.
Hence, the third angles of ∆ABC and ∆CAD are congruent
⟹∠CAB = ∠CDA
Using AAA rule, we can say, ∆ABC ~ ∆CAD
(AAA rule, which states if 3 angles are congruent between two triangles then the triangles are similar i.e. of the same shape which typically means the ratio of their areas, ratios of corresponding angles are equal)
Also, the ratio of areas of similar triangles is equal to the ratio of squares of corresponding sides.
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{{\text{A}}{{\text{D}}^2}}}{{{\text{A}}{{\text{B}}^2}}}$
From given AD = 4cm, AB = 5 cm
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{{4^2}}}{{{5^2}}}$
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{16}}{{25}}$
Note: In order to solve these types of questions the key is to make a precise figure which depicts all the given data. Then observe for similarities using properties of triangles. Keep in mind the AAA rule only states that the triangles are similar, they may or may not be congruent.

Complete step-by-step answer:
Given data,
∠ABD = ∠CAD. AB = 5cm, AD = 4cm and AC = 3cm.
Let us compare ∆ABC and ∆CAD
∠ABD = ∠CAD (given)
∠ACB = ∠ACD (common angle, from the figure)
The Third Angle theorem states, if two angles of a triangle are congruent to two angles in another triangle, then the third pair of angles are also congruent.
Hence, the third angles of ∆ABC and ∆CAD are congruent
⟹∠CAB = ∠CDA
Using AAA rule, we can say, ∆ABC ~ ∆CAD
(AAA rule, which states if 3 angles are congruent between two triangles then the triangles are similar i.e. of the same shape which typically means the ratio of their areas, ratios of corresponding angles are equal)
Also, the ratio of areas of similar triangles is equal to the ratio of squares of corresponding sides.
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{{\text{A}}{{\text{D}}^2}}}{{{\text{A}}{{\text{B}}^2}}}$
From given AD = 4cm, AB = 5 cm
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{{4^2}}}{{{5^2}}}$
⟹ $\dfrac{{{\text{A}}\left( {\Delta {\text{ACD}}} \right)}}{{{\text{A}}\left( {\Delta {\text{BCA}}} \right)}} = {\text{ }}\dfrac{{16}}{{25}}$
Note: In order to solve these types of questions the key is to make a precise figure which depicts all the given data. Then observe for similarities using properties of triangles. Keep in mind the AAA rule only states that the triangles are similar, they may or may not be congruent.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
