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Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ & QR and median PM of triangle PQR.Show that triangle ABC~∆PQR.?
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Sides AB and BC and median AD of a triangle ABC are respectively propo...
And QR and median PM of a triangle PQR. To prove that triangle ABC is similar to triangle PQR, we need to show that the corresponding angles are equal.
Let's start by labeling the given points and segments:
- Let A, B, C be the vertices of triangle ABC.
- Let P, Q, R be the vertices of triangle PQR.
- Let AB = x, BC = y, AD = m be the given side lengths of triangle ABC.
- Let PQ = kx, QR = ky, PM = km be the proportional side lengths of triangle PQR.
Since sides AB and BC are proportional to sides PQ and QR, we can write the following equations:
PQ/AB = QR/BC
kx/x = ky/y
k = y/x (equation 1)
Since median AD is proportional to median PM, we can write the following equation:
PM/AD = km/m
m = PM/AD (equation 2)
To prove that triangle ABC is similar to triangle PQR, we need to show that the corresponding angles are equal. Let's consider angle A in triangle ABC and angle P in triangle PQR.
Using the property of medians, we know that median AD divides triangle ABC into two triangles with equal areas. Similarly, median PM divides triangle PQR into two triangles with equal areas.
Therefore, we can say that the ratio of the areas of triangle ABC and triangle PQR is equal to the ratio of their corresponding medians squared:
Area(ABC)/Area(PQR) = (AD^2)/(PM^2) (equation 3)
Using the formula for the area of a triangle, we can express the areas of triangle ABC and triangle PQR in terms of their sides:
Area(ABC) = (1/2) * AB * AD
Area(PQR) = (1/2) * PQ * PM
Substituting the given proportional side lengths and the relationship between m and PM from equation 2, we have:
(1/2) * x * m / (1/2) * kx * km = (AD^2)/(PM^2)
m / (k * km) = (AD^2)/(PM^2)
m / (k^2 * m^2) = (AD^2)/(PM^2)
1 / (k^2 * m) = (AD^2)/(PM^2)
1 / (k^2 * m) = 1 (since the areas are equal, equation 3)
k^2 * m = 1
Substituting the relationship between k and x from equation 1, we have:
(y/x)^2 * m = 1
y^2 * m = x^2
Since y^2 * m = x^2, we can conclude that the corresponding sides of triangle ABC and triangle PQR are proportional.
Therefore, triangle ABC is similar to triangle PQR.
Let's start by labeling the given points and segments:
- Let A, B, C be the vertices of triangle ABC.
- Let P, Q, R be the vertices of triangle PQR.
- Let AB = x, BC = y, AD = m be the given side lengths of triangle ABC.
- Let PQ = kx, QR = ky, PM = km be the proportional side lengths of triangle PQR.
Since sides AB and BC are proportional to sides PQ and QR, we can write the following equations:
PQ/AB = QR/BC
kx/x = ky/y
k = y/x (equation 1)
Since median AD is proportional to median PM, we can write the following equation:
PM/AD = km/m
m = PM/AD (equation 2)
To prove that triangle ABC is similar to triangle PQR, we need to show that the corresponding angles are equal. Let's consider angle A in triangle ABC and angle P in triangle PQR.
Using the property of medians, we know that median AD divides triangle ABC into two triangles with equal areas. Similarly, median PM divides triangle PQR into two triangles with equal areas.
Therefore, we can say that the ratio of the areas of triangle ABC and triangle PQR is equal to the ratio of their corresponding medians squared:
Area(ABC)/Area(PQR) = (AD^2)/(PM^2) (equation 3)
Using the formula for the area of a triangle, we can express the areas of triangle ABC and triangle PQR in terms of their sides:
Area(ABC) = (1/2) * AB * AD
Area(PQR) = (1/2) * PQ * PM
Substituting the given proportional side lengths and the relationship between m and PM from equation 2, we have:
(1/2) * x * m / (1/2) * kx * km = (AD^2)/(PM^2)
m / (k * km) = (AD^2)/(PM^2)
m / (k^2 * m^2) = (AD^2)/(PM^2)
1 / (k^2 * m) = (AD^2)/(PM^2)
1 / (k^2 * m) = 1 (since the areas are equal, equation 3)
k^2 * m = 1
Substituting the relationship between k and x from equation 1, we have:
(y/x)^2 * m = 1
y^2 * m = x^2
Since y^2 * m = x^2, we can conclude that the corresponding sides of triangle ABC and triangle PQR are proportional.
Therefore, triangle ABC is similar to triangle PQR.
Community Answer
Sides AB and BC and median AD of a triangle ABC are respectively propo...
Given: Two triangles ΔABC and ΔPQR in which AD and PM are medians such that AB/PQ = AC/PR = AD/PM
To Prove: ΔABC ~ ΔPQR
Construction: Produce AD to E so that AD = DE. Join CE, Similarly produce PM to N such that PM = MN, also Join RN.
Proof: In ΔABD and ΔCDE, we have
AD = DE [By Construction]
BD = DC [∴ AP is the median]
and, ∠ADB = ∠CDE [Vertically opp. angles]
∴ ΔABD ≅ ΔCDE [By SAS criterion of congruence]
⇒ AB = CE [CPCT] ...(i)
Also, in ΔPQM and ΔMNR, we have
PM = MN [By Construction]
QM = MR [∴ PM is the median]
and, ∠PMQ = ∠NMR [Vertically opposite angles]
∴ ΔPQM = ΔMNR [By SAS criterion of congruence]
⇒ PQ = RN [CPCT] ...(ii)
Now, AB/PQ = AC/PR = AD/PM
⇒ CE/RN = AC/PR = AD/PM ...[From (i) and(ii)]
⇒ CE/RN = AC/PR = 2AD/2PM
⇒ CE/RN = AC/PR = AE/PN [∴ 2AD = AE and 2PM = PN]
∴ ΔACE ~ ΔPRN [By SSS similarity criterion]
Therefore, ∠2 = ∠4
Similarly, ∠1 = ∠3
∴ ∠1 + ∠2 = ∠3 + ∠4
⇒ ∠A = ∠P ...(iii)
Now, In ΔABC and ΔPQR, we have
AB/PQ = AC/PR (Given)
∠A = ∠P [From (iii)]
∴ ΔABC ~ ΔPQR [By SAS similarity criterion]
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Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ & QR and median PM of triangle PQR.Show that triangle ABC~∆PQR.?
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Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ & QR and median PM of triangle PQR.Show that triangle ABC~∆PQR.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ & QR and median PM of triangle PQR.Show that triangle ABC~∆PQR.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ & QR and median PM of triangle PQR.Show that triangle ABC~∆PQR.?.
Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ & QR and median PM of triangle PQR.Show that triangle ABC~∆PQR.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ & QR and median PM of triangle PQR.Show that triangle ABC~∆PQR.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ & QR and median PM of triangle PQR.Show that triangle ABC~∆PQR.?.
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