Answer:
68.3 cm
Step-by-step explanation:
To find the length of BH in the cuboid, let's break down the problem by step.
Given:
- BC = 27 cm
- CD = 54 cm
- DH = 32 cm
We need to find the length of BH. We'll use the Pythagorean theorem in two steps, as BH is the hypotenuse of the right-angled triangle BHC, and CH is the hypotenuse of the right-angled triangle CHD.
Calculate CH
In the right-angled triangle CHD:
- CH is the hypotenuse
- CD and DH are the legs
Using the Pythagorean theorem:
[tex]\sf CH^2 = CD^2 + DH^2 [/tex]
Substitute the known values:
[tex]\sf CH^2 = 54^2 + 32^2 [/tex]
[tex]\sf CH^2 = 2916 + 1024 [/tex]
[tex]\sf CH^2 = 3940 [/tex]
So,
[tex]\sf CH = \sqrt{3940} \, \textsf{cm} [/tex]
Calculate BH
In the right-angled triangle BHC:
- BH is the hypotenuse
- BC and CH are the legs
Using the Pythagorean theorem:
[tex]\sf BH^2 = BC^2 + CH^2 [/tex]
Substitute the known values:
[tex]\sf BH^2 = 27^2 + (\sqrt{3940})^2 [/tex]
[tex]\sf BH^2 = 27^2 + 3940 [/tex]
[tex]\sf BH^2 = 729 + 3940 [/tex]
[tex]\sf BH^2 = 4669 [/tex]
[tex]\sf BH = \sqrt{4669} [/tex]
[tex]\sf BH = 68.330081223426 [/tex]
[tex]\sf BH = 68.3 \, \textsf{cm (in 1 d.p)} [/tex]
Therefore, the length of BH is 68.3 cm.
