Question

eden is cutting two triangular tiles for her bathroom. she needs the tiles to be congruent but is not sure she is cutting them that way. eden has ensured that one side of both tiles is congruent. which pair of sides would eden need to compare in order to make sure the triangles are congruent by hl? grid image with triangles ( a, b, c ) (right triangle at ( a )) and ( d, e, f ) (right triangle at ( e )), with lengths 3.2, etc. options: ( overline{ac} ) and ( overline{fd} ), ( overline{ab} ) and ( overline{ef} ), ( overline{bc} ) and ( overline{ef} ), ( overline{bc} ) and ( overline{fd} )

Explanation:

Step1: Recall HL Congruence

HL (Hypotenuse - Leg) congruence applies to right - angled triangles. For two right - angled triangles to be congruent by HL, the hypotenuse and one leg of one triangle must be congruent to the hypotenuse and the corresponding leg of the other triangle.

First, identify the right - angled triangles. Triangle \(ABC\) and triangle \(DEF\) (assuming the second triangle is \(DEF\) with right angle at \(E\) and \(A\)) are right - angled triangles. The hypotenuses of the right - angled triangles are \(BC\) and \(FD\) (since \(BC = 3.2\) and \(FD\) is the hypotenuse of the lower right - angled triangle, and we know one leg is already considered). The legs of the right - angled triangles: we need to check the hypotenuse and the leg.

We know that \(BC\) and \(FD\) are the hypotenuses (length 3.2 as given for \(BC\) and \(FD\) should be equal if we consider the grid). Also, for HL, we need to check the hypotenuse and a leg. Wait, let's re - examine the options:

• Option 1: \(\overline{AC}\) and \(\overline{FD}\): \(AC\) is a leg, \(FD\) is a hypotenuse. HL requires hypotenuse and leg, not leg and hypotenuse in this way.
• Option 2: \(\overline{AB}\) and \(\overline{EF}\): \(AB\) and \(EF\) are the legs (the vertical or horizontal sides of the right - angled triangles), but HL needs hypotenuse and leg.
• Option 3: \(\overline{BC}\) and \(\overline{EF}\): \(BC\) is hypotenuse, \(EF\) is leg. Not the correct pair for HL.
• Option 4: \(\overline{BC}\) and \(\overline{FD}\): \(BC\) and \(FD\) are the hypotenuses (given \(BC = 3.2\) and \(FD\) should be equal as per the grid, and if we consider the right - angled triangles, and we already have one leg congruent (from the problem statement that one side is congruent), then comparing the hypotenuses (\(BC\) and \(FD\)) along with the already congruent leg would satisfy HL. Wait, actually, in HL, for two right - triangles, if the hypotenuse and one leg are congruent, then the triangles are congruent. Since we know one leg is congruent (from the problem: "Eden has ensured that one side of both tiles is congruent"), then we need to compare the hypotenuses. The hypotenuses of the two right - triangles are \(BC\) and \(FD\). So the pair of sides to compare is \(\overline{BC}\) and \(\overline{FD}\).

Answer:

\(\boldsymbol{\overline{BC}}\) and \(\boldsymbol{\overline{FD}}\) (the last option: \(\overline{BC}\) and \(\overline{FD}\))