Question

the diagram shows the shape of a plot of land that maria will use for her garden. she needs to know the length of each side so she can buy fencing. what is the length of line segment fj? 18 feet 30 feet 36 feet 42 feet

Explanation:

Step1: Apply the Pythagorean theorem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse of a right - triangle and \(a\) and \(b\) are the other two sides. In the right - triangle formed (by extending lines and considering the differences in lengths), if we assume the vertical side of the right - triangle related to \(FJ\) is \(24\) ft and the hypotenuse \(FG = 30\) ft. Let the horizontal part of \(FJ\) (the base of the right - triangle) be \(x\). Then \(x=\sqrt{FG^{2}-GH^{2}}\) (since \(GH\) is one of the non - hypotenuse sides of the right - triangle formed in the figure). But actually, if we consider the entire \(FJ\):
We can use the fact that if we complete the right - triangle, let the length of \(FJ\) be \(l\). We know that if we consider the right - triangle with hypotenuse \(FG = 30\) ft and one side \(24\) ft (the vertical side in the figure). By the Pythagorean theorem, the other side (let's call it \(a\)) of the right - triangle is \(a=\sqrt{30^{2}-24^{2}}\).

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Then \(l=a + 24\) (by adding the non - triangle part of \(FJ\)). Wait, no, actually, if we consider the whole \(FJ\):
We can use the Pythagorean theorem in reverse. Let's assume the figure is such that we can apply the Pythagorean theorem. If we consider the right - triangle formed (by the properties of the figure, assume the vertical side is \(24\) ft and the hypotenuse is \(30\) ft for a part of the figure). But actually, using the formula for the length of \(FJ\):
We know that if we consider the right - triangle with hypotenuse \(FG = 30\) ft and one side (vertical) \(24\) ft. Let the base of that right - triangle be \(x\). Then \(x=\sqrt{30^{2}-24^{2}}=\sqrt{(30 + 24)(30 - 24)}=\sqrt{54\times6}=\sqrt{324}=18\). But \(FJ\) is composed of two parts: one part is \(x\) (from the right - triangle) and another part. Wait, no, actually, if we use the formula \(FJ=\sqrt{30^{2}+24^{2}}\) is wrong. Wait, no, looking at the figure (assuming it's a right - angled trapezoid - like figure with \(FG = 30\) ft, \(GH=24\) ft, \(HJ = 24\) ft). If we complete the right - triangle (by extending lines), we can use the Pythagorean theorem.
Let's use the formula: If we consider the right - triangle with hypotenuse \(FG = 30\) ft and one side (the vertical side of the non - rectangle part) \(24\) ft. Let the base of that right - triangle be \(a\). Then \(a=\sqrt{30^{2}-24^{2}}=\sqrt{900 - 576}=\sqrt{324} = 18\). Then \(FJ=a+24\) (no, wait, no, actually, if we consider the entire \(FJ\) as the sum of two parts: one part is the base of the right - triangle (\(a\)) and another part. Wait, no, actually, if we use the formula for the length of \(FJ\):
We know that \(FJ=\sqrt{30^{2}+(24 + 24)^{2}}\) is wrong. Wait, no, looking at the figure (assuming it's a right - angled trapezoid - like figure with \(FG = 30\) ft, \(GH = 24\) ft, \(HJ=24\) ft). If we drop a perpendicular from \(G\) to \(FJ\) (let the foot be \(K\)). Then \(GK = 24\) ft (same as \(HJ\)), \(FG = 30\) ft. Then \(FK=\sqrt{FG^{2}-GK^{2}}=\sqrt{30^{2}-24^{2}}=\sqrt{(30 + 24)(30 - 24)}=\sqrt{54\times6}=\sqrt{324}=18\). Then \(FJ=FK+KJ\). Since \(KJ = 24\) ft (same as \(GH\)), \(FJ=18 + 24=42\) ft.

Answer:

42 feet