Question

The diagram shows a 7cm x 6cm rectangle-based pyramid all the diagonal sides - TA TB TC and TD are length 10cm M is the midpoint of the rectangular base work out the height MT to one decimal place

Answer 1

Given:

The base is a rectangle of dimension 7 cm×6 cm.

The length of each diagonal side = 10 cm

To find the height of MT.

Formula

By Pythagoras theorem we get,

$h^{2} = l^{2} + b^{2}$

where, h be the hypotenuse

b be the base and

l be the height.

Now, in this diagram,

The perpendicular distance of M from BC = 3 cm

Slant height = 10 cm

Taking, b = 3 and h = 10 we get,

$10^{2} = l^{2} + 3^{2}$

or, $l^{2} = 10^2-3^2$

or, $l = \sqrt{100-9}$

or, $l = 9.5$

Hence,

The height of MT is 9.5 cm.

Answer 2

Height MT of the pyramid will be 8.9 cm

    Given in the question,

• Dimensions of the rectangular base = 7 cm × 6 cm
• M is the center of the rectangular base.
• Slant height of the pyramid = 10 cm

Since, diagonals of a rectangle bisect each other, point M will be the midpoint of the diagonal AC.

By applying Pythagoras theorem in right triangle ΔABC,

AC² = AB² + BC²

AC = $\sqrt{AB^2+BC^2}$

AC = $\sqrt{6^2+7^2}$

AC = $\sqrt{85}$

Therefore, AM = $\frac{\sqrt{85} }{2}$

AM = 4.61 cm

Apply Pythagoras theorem in right triangle ΔAMT,

AT² = AM² + TM²

TM = $\sqrt{AT^2-AM^2}$

TM = $\sqrt{(10)^2-(4.61)^2}$

     = $\sqrt{78.7479}$

     = 8.874

     ≈ 8.9 cm

    Therefore, height MT of the pyramid will be 8.9 cm.

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