Question

Triangle XYZ is an isosceles triangle with right angle at Y. If XZ=12cm,find the length of the altitude from Y to XZ

Answer 1

Answer

6 cm

Explanation

(Refer attachment for figure)

Given, XYZ is an isoceles triangle, right angled at Y.

XZ = 12 cm

To find,

The length of altitude from Y to XZ

Since, XYZ is an isoceles triangle,

⇒ XY = YZ

Let, XY = YZ = a

Then, applying Pythagoras Theorem, we get

a² + a² = 12²

⇒ 2a² = 12²

Taking root on both sides.

a√2 = 12

⇒ a = 12/√2

Rationalising,

⇒ a = 6√2

Now, area of triangle = 1/2 × base × height

⇒ area of triangle XYZ = 1/2 × XY × YZ

⇒ area of triangle XYZ = 1/2 × 6√2 × 6√2

⇒ area of triangle XYZ = 36 cm²

Let the altitude from Y to XZ be YM

In that case, area of triangle XYZ = 1/2 × XZ × YM

But we know that area of triangle = 36 cm²

⇒ 1/2 × XZ × YM = 36

⇒ 1/2 × 12 × YM = 36

⇒ 6 × YM = 36

⇒ YM = 6 cm  

Answer 2

Question :-- Triangle XYZ is an isosceles triangle with right angle at Y. If XZ=12cm,find the length of the altitude from Y to XZ ?

Solution (1) :--

❁❁ Refer To Image First .. ❁❁

Let , XY = YZ = a cm.

Using pythagoras Theoram in Rt∆XYZ now,

→ XY² + YZ² = XZ²

→ a² + a² = (12)²

→ 2a² = 144

→ a² = 72

→ a = 6√2 cm. = XY = YZ

Now, Area of Right ∆ = 1/2 * Base * Perpendicular = 1/2 * Hypotenuse * Length of altitude To Vertex.

Putting values and comparing their Area we get,

→ 1/2 * 6√2 * 6√2 = 1/2 * 12 * Altitude .

→ 36 * 2 = 12 * Altitude

→ Altitude = 72/12 = 6cm.

Hence, Length of the altitude from Y to XZ is 6cm.

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Solution (2) :--

After Finding the Sides of Right Angle ∆XYZ , we have now,

→ XY = YZ = 6√2 cm.

→ XZ = 12cm.

Let, XW = m cm.

Than, WZ = (12-m) cm. (As XZ is 12cm. )

Now, in Right Angle ∆ we know, That,

→ XY² = XW * XZ

Putting values here , we get,

→ (6√2)² = m * 12

→ m = 72/12 = 6cm.

So, WZ = (12-m) = 12-6 = 6 cm .

we have now, XW = WZ = 6 cm Each .

Again, in Right Angle ∆, we have ,

→ WY² = XW * WZ

putting values here now, we get,

→ WY² = 6 * 6

→ WY² = 6²

Square - root both sides ,

→ WY = 6 cm.

Hence, Length of the altitude from Y to XZ is 6cm.

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Solution (3) :---

when we Find XW = WZ = 6cm Above, with this we can say that,

→ Point W is the circumcentre of Right ∆XYZ. As we know that, circumcentre of Right angle ∆ divides Hypotenuse in 2 Equal Parts .

So,

→ XW , WZ and WY are circum-Radius of Circumcircle .

Hence, XW = WZ = WY = 6cm. Each .

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Equations are just the boring part of mathematics. I attempt to see things in terms of geometry.

This is the beauty of Geometry ..

If our Basic is Clear, we can Solve any problem with As many as Methods we want ..