Advertisements
Advertisements
Question
ABC is an isosceles triangle in which AC = BC. AD and BE are respectively two altitudes to sides BC and AC. Prove that AE = BD.
Advertisements
Solution
Given: ΔABC is an isosceles triangle in which AC = BC.
Also, AD and BE are two altitudes to sides BC and AC, respectively.
To prove: AE = BD
Proof: In ΔABC,
AC = BC ...[Given]
∠ABC = ∠CAB ...[Angles opposite to equal sides are equal]
i.e., ∠ABD = ∠EAB ...(i)
In ΔAEB and ΔBDA,
∠AEB = ∠ADB = 90° ...[Given, AD ⊥ BC and BE ⊥ AC]
∠EAB = ∠ABD ...[From equation (i)]
And AB = AB ...[Common side]
∴ ΔAEB ≅ ΔBDA ...[By AAS congruence rule]
⇒ AE = BD ...[By CPCT]
Hence proved.
APPEARS IN
RELATED QUESTIONS
ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.
BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.
In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent.
Prove that in a quadrilateral the sum of all the sides is greater than the sum of its diagonals.
In the following figure, BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that ∆ABC ≅ ∆DEF.
ABC and DBC are two triangles on the same base BC such that A and D lie on the opposite sides of BC, AB = AC and DB = DC. Show that AD is the perpendicular bisector of BC.
Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side.
Two lines l and m intersect at the point O and P is a point on a line n passing through the point O such that P is equidistant from l and m. Prove that n is the bisector of the angle formed by l and m.
ABCD is a quadrilateral such that diagonal AC bisects the angles A and C. Prove that AB = AD and CB = CD.
ABC is a right triangle such that AB = AC and bisector of angle C intersects the side AB at D. Prove that AC + AD = BC.
