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Find the height of triangle ABC

Mathematics

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Answer

2.55 in\text{2.55 in}

Step-by-step solution

4 steps

To find the height of triangle ABC, we first need to interpret the given information. We are given two side lengths, AC=4 inAC = 4 \text{ in} and BC=3.3BC = 3.3. The diagram shows a height drawn from vertex C perpendicular to the base AB. For this problem to be solvable with only two side lengths and no angles or the base length, we must assume that triangle ABC is a right-angled triangle with the right angle at C. The height we need to find is the altitude to the hypotenuse.

  1. Identify the given side lengths and assume a right angle.

    We have the lengths of the two legs of the right-angled triangle ABC:
    AC=4 inAC = 4 \text{ in}
    BC=3.3 inBC = 3.3 \text{ in}
    We assume ∠C=90∘\angle C = 90^\circ. Let hh be the height from C to the hypotenuse AB.

  2. Calculate the length of the hypotenuse AB.

    Using the Pythagorean theorem:

    AB2=AC2+BC2AB^2 = AC^2 + BC^2 AB2=(4 in)2+(3.3 in)2AB^2 = (4 \text{ in})^2 + (3.3 \text{ in})^2 AB2=16 in2+10.89 in2AB^2 = 16 \text{ in}^2 + 10.89 \text{ in}^2 AB2=26.89 in2AB^2 = 26.89 \text{ in}^2 AB=26.89 inAB = \sqrt{26.89} \text{ in}

  3. Calculate the height hh from C to AB.

    The area of a right-angled triangle can be expressed in two ways:
    Using the legs as base and height: Area=12×AC×BCArea = \frac{1}{2} \times AC \times BC
    Using the hypotenuse as base and hh as height: Area=12×AB×hArea = \frac{1}{2} \times AB \times h
    Equating these two expressions:

    12×AC×BC=12×AB×h\frac{1}{2} \times AC \times BC = \frac{1}{2} \times AB \times h h=AC×BCABh = \frac{AC \times BC}{AB}

    Substitute the values:

    h=4 in×3.3 in26.89 inh = \frac{4 \text{ in} \times 3.3 \text{ in}}{\sqrt{26.89} \text{ in}} h=13.226.89 inh = \frac{13.2}{\sqrt{26.89}} \text{ in}

  4. Compute the numerical value.

    h≈13.25.185556 inh \approx \frac{13.2}{5.185556} \text{ in} h≈2.54558 inh \approx 2.54558 \text{ in}

    Rounding to two decimal places:

    h≈2.55 inh \approx 2.55 \text{ in}

    The height of triangle ABC is approximately 2.55 in\text{2.55 in}.

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