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A regular tetrahedron is a three-dimensional shape with four faces, each of which is an equilateral triangle. All edges are of equal length, in this case, 4 cm.
ScanSolve AI Answer
Step-by-step solution
3 stepsTo solve this problem, we need to visualize the tetrahedron's net and then find the shortest path on the unfolded surface.
A regular tetrahedron is a three-dimensional shape with four faces, each of which is an equilateral triangle. All edges are of equal length, in this case, 4 cm.
The path from P to S passes over the edge QR at its midpoint T. This means the path lies on two faces: and . To find the straight path, we need to unfold these two faces into a single plane.
Draw the net of the relevant faces.
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Draw an equilateral triangle PQR with side length 4 cm.
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Draw another equilateral triangle SQR, sharing the edge QR, such that vertex S is on the opposite side of QR from vertex P. This forms a larger triangle (or a rhombus if P, Q, S, R were coplanar, but here P and S are vertices of two triangles sharing a base).
The net will look like this:
P / \ / \ / \ Q-------R \ / \ / \ / S
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Locate the midpoint T and draw the path.
- Mark the midpoint T on the edge QR.
- The straight path from P to S, passing through T, is a straight line segment connecting P and S on this unfolded net.
Calculate the length of the path PS.
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The line segment PT is the altitude (height) of the equilateral triangle PQR from vertex P to the base QR. The formula for the altitude of an equilateral triangle with side length is . For , cm. So, the length of cm.
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Similarly, the line segment ST is the altitude of the equilateral triangle SQR from vertex S to the base QR. For , cm. So, the length of cm.
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On the unfolded net, P, T, and S are collinear. Therefore, the total length of the straight path PS is the sum of PT and ST. Length of path PS = .
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To get a numerical value, we use . Length of path PS cm.
The length of the straight path of PS through the midpoint T over the edge QR is or approximately .
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